Using conceptual metaphor and functional grammar to explore how language used in
physics affects student learning
David T. Brookes
Department of Physics; Loomis Laboratory of Physics; 1110 West Green St.; Urbana, IL 61801-3080
Eugenia Etkina
The Graduate School of Education; 10 Seminary Place; New Brunswick, NJ 08901
This paper introduces a theory about the role of language in learning physics. The theory is
developed in the context of physics students’ and physicists’ talking and writing about the subject
of quantum mechanics. We found that physicists’ language encodes different varieties of analogi-
cal models through the use of grammar and conceptual metaphor. We hypothesize that students
categorize concepts into ontological categories based on the grammatical structure of physicists’
language. We also hypothesize that students over-extend and misapply conceptual metaphors in
physicists’ speech and writing. Using our theory, we will show how, in some cases, we can explain
student difficulties in quantum mechanics as difficulties with language.
PACS numbers: 01.40.Fk;01.40.Ha;03.65.-w
I. INTRODUCTION
A. Our Starting Point
The goal of this paper is to present a theoretical frame-
work explaining the role of spoken and written language
in physics. This framework can be used to probe how
physicists represent their ideas in language and more im-
portantly, to understand how physics students interpret
language they read and hear. We will use the frame-
work to understand the types of meaning students may
construct from language and the sorts of difficulties they
may encounter in trying to construct meaning from the
language that they read and hear in physics. We are
going to suggest that there are some student difficulties
that may be recognized primarily as difficulties with lan-
guage. Below we present two initial theoretical points
that will help the reader understand the role of language
in learning and communicating physics.
1. Language as a Representation
We will adopt Jay Lemke’s view that the primary ac-
tivity that students encounter and participate in, in a
physics course, is representing [1]. They encounter many
different representations of physics ideas: graphs, equa-
tions, tables, pictures, diagrams, and words. These repre-
sentations of physics ideas are each by themselves incom-
plete. It takes an act of assimilating, coordinating, and
moving between many different representations in order
to create understanding. Therefore one of the first abil-
ities students have to develop is the ability to represent
ideas and physical processes in different ways and move
between representations. Physicists are conscious of the
role of equations and graphs in their reasoning. Less
attention, however, has been paid to language as a repre-
sentation of knowledge and ideas in physics. Our starting
point will be to treat language as a legitimate representa-
tion of physical ideas and processes. Physicists are aware
that some student difficulties may be caused by confusing
language (see for example, [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]),
but only a relatively small amount of research has been
done in this area [12, 13, 14].
2. Information and Communication
Reddy [15] has suggested that people construct mean-
ing from the words that they hear, based on their prior
knowledge and experience. For example, If you ask some-
one: “Are you sad?” And they respond with a “1”: What
would this response mean? By itself it means nothing,
it is simply a signal. Imagine that you and your partner
have a list of possible responses: either 1, 2 or 3. This
is called a “repertoire” of responses. After the two of
you have established a repertoire of responses you need
to assign a meaning to them. Say you two agree before
hand that 1 = “yes,” 2 = “no,” and 3 = “unable to
give a definite answer.” Now you both have established
a shared repertoire of acceptable signals, plus a shared
code. You and your partner have the means to com-
municate. This example shows that by itself the signal
is meaningless. A recipient has to construct the mean-
ing using a commonly understood repertoire and a pre-
viously shared code (shared a priori between sender and
receiver).
From the above discussion, it follows that meaning can-
not be directly passed, conveyed or in any way trans-
ported from the instructor to the student. The teacher
has to help the student construct meaning by elaborat-
ing the code. Students can then use this code to decode
the words that the instructor uses. For example, when a
physicist says “the electron is in the ground state,” she
means that the electron has a particular energy. How-
ever, if the students do not share the code for the word
http://arXiv.org/abs/0704.1319v1
“state” as the energy state, they may construct a spatial
interpretation from the same statement.
B. Overview of the paper
In Section II we will explain our theoretical frame-
work in the context of the data we gathered. First,
we will describe our language data sources (QM text-
books, interviews with physics professors, and videos of
QM students working on QM problems.). To be able
to explain how language works in physics, we found it
necessary to introduce the formal categories of analogy,
metaphor, grammar and ontology. We will elaborate how
analogy, metaphor, grammar and ontology fit together to
describe physicists’ language when they “speak and write
physics”. Finally, we will present two hypotheses about
how one can use our theoretical framework to understand
how students are interpreting the language that they read
and hear in a physics class.
In Section III we will return to the interviews with
physics professors and the QM textbooks and code the
language used. By looking for patterns of usage that can
be described and explained by the theoretical framework
we have developed, we will show how this framework is
applicable for understanding how physicists use language
in their reasoning process.
In Section IV we will describe how we tested the ap-
plicability of our theoretical framework for understand-
ing students’ reasoning and learning in physics. We will
present two case studies from our video data of physics
students working on QM problems.
In Section VI we will explore some future directions
that this research on language in physics could proceed.
II. THEORETICAL FRAMEWORK
A. Introduction
Our theory was developed from a number of sources of
data: (1) Interviews with 5 physics professors. (2) Orig-
inal QM papers from Born [16] and Schrödinger [17], as
well as an analysis from Goldstein [18] of how Schrödinger
developed the wave equation. (3) A selection of older and
more modern, popular introductory quantum mechanics
textbooks [19, 20, 21, 22, 23, 24]. (4) Two physics stu-
dent homework study groups.
We began by comparing the way Schödinger and Born
wrote about their ideas with the way modern textbooks
and physics professors wrote and spoke about the same
ideas. This led us to define two separate patterns of
language used to express ideas in QM:
1. The first pattern was language used by the in-
ventors of QM. They tended to use cautious and
figurative language. Ideas were often expressed
as comparisons of the form “X is like Y in cer-
tain respects.” They made analogies explicit and
cautioned against overextending or misinterpreting
these analogies.
2. The second pattern we observed was language used
to communicate already established knowledge of
QM. (Language used by physics professors and
modern QM textbooks.) This language was char-
acterized by statements of fact with little if any ref-
erence to the original analogies on which the ideas
were based.
These two patterns of language lead us to investigate the
role of analogy and metaphor in describing physicists’
language.
In our data of students working on their QM home-
work problems, we focussed our attention on episodes
when students stopped calculating, and engaged in an
activity that could loosely be described as sense-making.
In these episodes it appeared to us as if students under-
stood the physical ideas, but they were confused about
the language used to express the physical ideas. We hy-
pothesized that students were confused by the figurative
language that physicists used to describe their ideas. We
will justify this claim in Section IV.
B. Analogical Models Encoded as Metaphors in
Physics
1. Metaphors in Physics Language
Lakoff and Johnson [25] have hypothesized that hu-
man language and the human conceptual system are
largely made up of unconscious conceptual metaphors.
We have extended this idea to physics by suggesting that
physicists speak and write using conceptual metaphors.
For example, physicists talk about “diffraction of elec-
trons” and a “wave equation for the electron.” Both
phrases suggest the conceptual metaphor the electron
is a wave. Conceptual metaphors are often unconscious
metaphors and seldom made explicit. They have become
quite literal, losing their figurative origin through their
unconscious and frequent use. For a more complete dis-
cussion of what a metaphor is and how it differs from
analogy and simile, we refer the interested reader to [26].
Other excellent discussions of the theoretical status of,
and issues surrounding, metaphor and analogy may be
found in [27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
2. Types of Analogies Encoded as Metaphors
Researchers have shown that physicists use analogical
models to construct new ideas [37, 38]. These analogical
models become, in time, encoded linguistically as concep-
tual metaphors [39, 40]. The way physicists talk about
already established knowledge is different than the way
they talk about new ideas they are trying to comprehend
themselves.
We will take this idea further. From the primary data
(textbooks, original papers, and interviews with physics
professors), we have identified three types of analogical
model that metaphors encode. These can be classified by
their origin and function:
1. Current analogical models: For example,
Schrödinger based his wave equation on an anal-
ogy to wave optics [17, 18]. The corresponding
metaphorical system is the electron is a wave
and is spoken about by modern physicists in terms
such as “electron interference,” “electron diffrac-
tion,” “wave equation,” and so on.
2. Defunct analogical models: It is often the case
in physics that older models, whose limitations
have been experimentally exposed and supplanted
by better models, live on in the language of physics.
The caloric theory of heat lives on in phrases which
reflect the heat is a fluid metaphor. For exam-
ple, “heat flows from object A to object B.” Physi-
cists use these metaphorical pictures when they rea-
son. We will elaborate this point further below.
3. Descriptive analogies: For example an anal-
ogy between a physical valley and a potential en-
ergy graph. The metaphor is potential energy
graphs are water wells. Examples of how the
metaphor is used in language are: “potential well,”
“potential step,” “energy level,” “ground state,”
and so on.
We will identify metaphors that encode analogies 1 to
3 by identifying the base of the analogy [35]. We will use
the idea that conceptual metaphors borrow terms from
the base of the analogy and apply these words directly to
the target concept. For example, if we look at the matter-
wave analogy in QM, we can consider that a water wave
or an electro-magnetic wave is the prototypical example
which will serve as the “base” of the analogy. Thus words
such as “interfere,” “polarize,” “diffract,” and “wave” are
used in the context of “an electron.” Such examples will
be identified as instances of the electron is a wave
metaphor.
3. Features and Functions of Metaphors in Physics
We hypothesize that physicists unconsciously prefer to
speak and write in metaphors because metaphors have
certain features and functions that are advantageous to
them. The features and functions of these metaphorical
systems are listed below with examples from interview
data with physics professors.
Feature 1: Conceptual metaphors encode analogies.
They encode a more deep and complex piece of knowledge
which is the completely elaborated analogy. That elabo-
ration as an analogical model is, however, tacit amongst
the community who use the metaphor and associated
model regularly.
Function: Physicists are able to use these metaphor-
ical systems to reason productively about a particular
situation or problem. For example, the electron is a
wave metaphor can be used productively to explain the
Heisenberg uncertainty principle:
“I often think of it. . . in terms of Fourier
transforms and the reciprocity between the
bandwidth of the channel and the length of
the signal pulse that can be detected.” (Prof
Note the use of words from the base domain of electro-
magnetic waves: “Fourier transforms,” “bandwith,” and
“signal pulse” in particular.
Even defunct analogies (type 2) represent productive
modes of thought for physicists. There is a class of prob-
lems for which it is quite adequate to talk about heat
as a fluid. For example, when there is no work being
done on or by the thermodynamic system, it is satisfac-
tory to think of heat flowing into or out of the system and
that the change in temperature of the system is directly
proportional to the amount of heat gained or lost.
Feature 2: Metaphorical systems are partial in na-
ture. This means that more than one metaphorical sys-
tem is needed to fully understand a physical concept.
Function: We observed that physicists switch easily
and unconsciously between one system and another de-
pending on the type of question that is asked. For exam-
ple, in the following extract, Prof. D switches back and
forth between particle and wave metaphor to describe the
process of electrons passing through a Young’s double slit
apparatus.
“Of course in any one experiment,. . . you will
not observe. . . an interference pattern on the
screen [wave metaphor] — if all you do is to
scatter one electron [particle metaphor]. The
intensities are just too low [wave metaphor].
. . . you have to have a large number of elec-
trons [particle metaphor], you have to have a
beam of electrons [wave metaphor]. And each
electron will contribute a little piece of the in-
tensity that you see on that screen [particle
metaphor]. What I envisage is. . . a beam of
electrons which can be represented by a plane
wave [wave metaphor]. . . ”
Feature 3: Metaphors involve the use of the verb “is”
rather than “is like.” Metaphors are grammatically iden-
tifying relational processes, i.e., they are grammatically
equivalent to statements of category membership.
Function: We hypothesize that metaphor reflects a par-
ticular aspect of an expert physicist’s thought process.
The use of metaphor itself rather than simile is signifi-
cant. Irrespective of deep philosophical discussions about
what is “real,” it seems apparent that physicists them-
selves need to assert something stronger than “like” —
they need to assert “is” in their own reasoning process.
We suggest that this is a fundamental trait of how knowl-
edge is generated and represented in physics. It is signifi-
cant because such assertions may often conceal the vague
or partial nature of metaphor itself.
For example, Prof. D provided the following response
to the question: What happens to a single electron when
it passes through a Young’s double slit apparatus?
“. . . [to] understand that experiment, you’ve
got to forget about the idea that an electron is
a particle. It is not a particle in that context,
it behaves like a wave. So you just think of
it as a plane wave [our emphasis] advancing
on the two slits, and the interference between
the two. . . outgoing beams, just using Huy-
gen’s principle, leads to the. . . interference
pattern. . . ”
Note that comparison, “it behaves like a wave,” is fol-
lowed directly by, “just think of it as a plane wave.”
Feature 4: The apparatus of language constrains the
ways physicists can talk about physical phenomena and
therefore constrains the types of models that can be rep-
resented in language.
Function: Descriptive analogies (type 3) encoded
as metaphors also represent ways of speaking
about/describing physical systems. This is very
important because there is a limit on what can be
represented with language. Such metaphors also give
abstract concepts and quantities a grounding in physical
reality and physical experience.
Consider for example, the modern physicist’s view of
energy. Physicists can define energy as a state function
yet can physicists speak literally about energy as a state
function? Our hypothesis is that it is simply impossible
to come up with grammatical constructions that convey
the meaning of energy as a state function. The very best
locutions are “energy flows into the system,” or “process
X caused the kinetic energy of the system to increase.”
In both these cases, metaphorically, energy is being spo-
ken of as matter and the system as a container of energy.
(This is suggested particularly by the use of the preposi-
tions “into” and “of.”) It is no coincidence that these two
locutions are identical to examples given by Lakoff and
Johnson [25]. The authors describe similar metaphori-
cal patterns in how humans (in English at least) encode
physical processes and events as movement of substances
into and out of containers.
Physicists are aware of the limitations of their lan-
guage. When asked about what is oscillating in a quan-
tum mechanical wave, one professor responded:
Prof B: “The problem is you’re trying to shoe-
horn a phenomenon into ordinary everyday
English language, and I think the problem is
with the language, not with the phenomenon.
So, if you ask me to explain it in English, I
think English has limitations which make it
impossible to give a satisfactory explanation
in English. But, I don’t have to understand
it in English. I mean, I think I sort of know
what’s going on. At least I have realized the
limitations in English and, it doesn’t bother
C. Ontological Underpinnings
1. Introduction
It has been suggested that humans divide the world
into ontological categories of matter, processes and men-
tal states [41]. In this section we will show that this
idea can be applied to models in physics. The ele-
ments of a physical model: the objects or systems of
objects, interaction laws, force laws, state laws etc.,
may be mapped to the ontological categories of matter,
processes, and physical states. In cognitive linguistics,
Lakoff and Johnson [25] have shown that systems of con-
ceptual metaphors are based on ontological metaphors.
These ontological metaphors often give abstract concepts
an existence as concrete objects or things. To unite these
two views and systematize our linguistic analysis, we hy-
pothesize that ontological metaphors in physics language
are realized as grammatical metaphors. Functional gram-
marians have suggested [42] that the elements of a sen-
tence can be divided into participants (nouns or noun
groups), processes (verbs or verb groups), and circum-
stances (generally adverbial or prepositional phrases). In
order to unify the metaphorical and grammatical views,
we have suggested [26, 43] that grammatical partici-
pants should be mapped to the ontological category of
matter, and grammatical processes represent ontological
processes. Ontological physical states also have unique
grammatical representations, through the use of gram-
matical location.
2. A Lexical Ontology
We hypothesize that the concepts in a physical or ana-
logical model can be arranged into an ontological tree
similar to the one proposed by Chi et al. [41]. It is nec-
essary to modify Chi et al.’s ontology tree to accommo-
date one missing category: namely physical states. (See
Fig. 1.)
Etkina et al. have suggested that physical models can
be broken up into a taxonomy of (1) models of objects,
(2) models of interactions between objects, (3) models
of systems of objects, and (4) models of processes that
the objects/system undergoes [44]. In addition to their
taxonomy, we are going to suggest that there are two
classes of physical variables that describe a system or
the objects in it. These are (5) physical properties of
FIG. 1: A revised ontology tree based on [41]
objects (such as mass and charge), and (6) state vari-
ables that describe a configuration of the system (e.g.,
position, momentum) or state functions defined over a
system configuration (e.g., energy, entropy).
We will now show how Etkina et al.’s model taxonomy
can be mapped into the ontology tree shown in Fig. 1.
This mapping is shown in Table I.
Physical properties such as mass and charge should be
considered properties of objects classified in the matter
category.
The categorization of concepts in physics into an on-
tology tree (as shown in Table I), will be termed a lexical
ontology. For example, physicists generally agree that
energy is a state function, while heat and work are pro-
cesses by which energy is transfered into or out of a sys-
tem. Thus a lexical ontology refers definitions of physics
concepts into matter, processes, and states that physicists
would agree with as a community.
3. Grammar and Ontology
Although physicists can agree on the meaning of terms,
how do they represent the ontology of physics concepts
with language?
We suggest that every physical model described in lan-
guage has an ontology and that this ontology is encoded
in the grammar of the sentence. This grammatical on-
tology can be either literal or figurative (metaphorical).
If the lexical ontology matches the grammatical ontology
then the sentence is literal. If the lexical ontology does
not match the grammatical ontology of the same term in a
given sentence, then a grammatical metaphor is present.
We suggest that these metaphors may be consistently
identified by using the grammatical/ontological analysis
elaborated below. For an introduction to the methods of
functional grammar, we refer the reader to [42].
Consider for example, “John [agent] kicked [process]
the ball [medium].” Here “John” and “the ball” are gram-
matical participants, functioning grammatically as ob-
jects or matter. We also recognize that “John” and “the
ball” are naturally defined as matter in some sense. Thus
the grammatical ontology and lexical ontology match.
There is nothing metaphorical in this sentence. Consider
now for example, “heat [medium] flows [process] from the
environment to the gas.” In this sentence a physicist
would recognize heat to define a process of movement of
energy into the system (lexical ontology). But grammat-
ically “heat” is functioning as a participant, namely heat
is the matter that is flowing. In this case the grammati-
cal function of the term “heat” and the lexical ontology of
“heat” contradict each other. The sentence is therefore
metaphorical.
We are going to propose the following mapping from
grammar to the ontology tree shown in Fig. 1: Gram-
matical participants should be mapped into the ontolog-
ical category of matter. Participants can immediately be
separated into living and non-living ontological subcate-
gories: Beneficiary, agent, and medium (as it participates
in an action process, such as “a force [medium] acts [pro-
cess]”), can all be thought of as living entities. Range and
medium (as it participates passively in an event process
such as “heat [medium] flows [process]”) can be thought
of as non-living entities.
Certain parts of circumstantial elements can also be
mapped to the matter category. In the example, “. . . the
incident particles [medium] will be. . . partially transmit-
ted [process] through the potential-well region [location].”
“the potential-well region” could be classified as non-
living matter. However location also functions grammati-
cally to make ontological physical states as in “A particle
[medium] is [relational process] at coordinates (1,1,1) [lo-
cation].” This will be discussed further below.
An important type of grammatical process in the dis-
course of physics is the relational process. Relational
processes are processes of being in that they almost al-
ways include some form of the verb “to be.” Relational
processes have two modes: identifying and attributional.
The identifying mode is a reflexive relationship. For ex-
ample, “the neutrino is the lightest known particle.” It
makes sense to say “the lightest known particle is the neu-
trino.” The attributional mode denotes category mem-
bership and is not reflexive. For example: “An electron
is a lepton.”
We hypothesize that physical states (as expressed in
physicists’ language) are commonly comprised of identi-
fying relational processes where the second identifier is
missing and replaced by a grammatical circumstance of
location. Typical examples are: “The electron is in the
ground state,” “the particle is at such and such coordi-
nates.” Such sentences very often involve a grammatical
TABLE I: Table illustrating how Etkina’s model taxonomy successfully maps into Chi’s (modified) ontology
Ontological
category
Matter Process State
Ontological sub-
category
Non-living Event Procedure and Constraint-
based Interaction
Physical State
Taxonomy
element
objects system interaction
causal laws, state laws state variables, state
functions
metaphor. Ontologically location is mapped to some sort
of physical object or matter, this often conflicts with the
lexical ontology. These grammatical metaphors corre-
spond directly to the ontological metaphors of Lakoff and
Johnson in choice of preposition: “in” implies container,
“at” implies point location in either time or space, “on”
implies surface. We believe that it is also no coincidence
that these statements have a grammatical structure iden-
tical to those of mental states. For example, in English
we say, “I am in love,” or “I am in trouble,” or “I am in
a state of confusion” etc. It seems to us that physicists
have borrowed this metaphor wholesale and blended it
with the notion of a physical state, to create a way of
speaking about physical states.
Ontological processes that describe the behavior of a
physical system, are realized in speech by grammatical
material processes. Relational processes realize either
physical states as shown above, or denote some compo-
nent of the model in the sense of category membership.
In grammar there are two types of material process: ac-
tion and event. These two types of process can be used
to distinguish between living and non-living matter.
The entire mapping from grammar to ontological cat-
egory is summarized in Table II below.
TABLE II: Summary of the mapping between grammar and
ontological category
Grammatical
function
Ontological category
(If X functions grammat-
ically as. . . )
−→ (. . . classify X ontologi-
cally as. . . )
Agent −→ Matter:living
Beneficiary −→ Matter:living
Medium (action process) −→ Matter:living
Medium (event process) −→ Matter:non-living
Role −→ Matter:non-living
Objects in Location −→ Matter:non-living
Process −→ Process
Manner −→ Process
D. Summary
The theoretical framework is summarized in Fig. 2 be-
Consider, for example, the caloric theory of heat. This
theory of thermodynamics began as an analogy to a
FIG. 2: Summary of the role of analogy, metaphor, ontology
and grammar
weightless fluid in the late eighteenth century. Over time
the elements of this theory became encoded in the lan-
guage of physics as a conceptual metaphor. For example,
physicists today still say “heat flows from object A to ob-
ject B,” and talk about the “heat capacity” of an object.
Phrases and sentences such as these are evidence of the
conceptual metaphor heat is a fluid in physicists’ lan-
guage. For physicists, speaking about heat is a fluid is a
productive mode of reasoning as long as there is no work
being done on or by the thermodynamic system. The ap-
plicability and limitations of talking about heat as a fluid
are communally well understood. The analogy between
heat and a fluid has an underlying ontology of matter
(the heat fluid), processes (the movement of heat from
one object to another), and states (the amount of heat
in an object — indicated by the object’s temperature).
This ontology is encoded in the grammar of each sen-
tence used to speak or write about the thermodynamic
system. In the example, “heat flows from object A to
object B,” heat is a grammatical participant, while the
grammatical process is “flows”. Object A and B are parts
of grammatical location. Implicitly, the amount of heat
in object A or object B indicates the current state of the
system. In the modern thermodynamic model, the onto-
logical matter is the atoms or molecules in the system,
the processes that the system undergoes are heating and
work (processes of energy transfer), and the state of the
system is represented by the energy or entropy of the sys-
tem for a given configuration of the molecules. Note how
the modern ontology is in direct conflict with the caloric
ontology of thermodynamics. Speaking about heat as
matter is therefore a grammatical metaphor. It is gram-
matical metaphors like this that underpin the conceptual
metaphor heat is a fluid.
E. Student Difficulties, Student Learning
The central question of our paper is: What is the in-
terplay between the linguistic representations that physi-
cists use and students’ learning and students’ difficulties?
We will narrow this down to two hypotheses regarding
the role of language and learning in physics. These are
elaborated in Sections II E 1 and II E 2 below.
1. Student Difficulties Interpreting Metaphors
Students struggle to see the applicability and limita-
tions of analogies that they encounter. We suggest the
same applies to metaphorical language that they hear
and read. To comprehend a metaphor people construct
an ad hoc category [32, 33]. This means that a state-
ment of the form “X is Y” has to be interpreted through
the formation of a new shared category (an ad hoc cate-
gory) of which Y is a prototypical member. For example,
to comprehend a metaphor such as the electron is
a smeared paste, the reader has to come up with an
ad hoc category shared by both entities. A physicist who
understands the quantum mechanical behavior of an elec-
tron, might suggest an ad hoc category of “things that
don’t have a well-defined location.” There is no guaran-
tee that a student will come up with the same classifica-
tion. We hypothesize that students often come up with
an ad hoc category that is inappropriate to a given situa-
tion. This inappropriate categorization is at the heart of
their difficulties. These difficulties may manifest them-
selves as “misconceptions” or student difficulties. We
predict that students will overextend and misapply key
aspects of metaphorical systems in physics. Instances
where metaphors are overextended or taken too literally
will be connected with their faulty reasoning.
In order to test these ideas it is first necessary to iden-
tify if there are really coherent systems of conceptual
metaphors in the way physicists speak and write. In Sec-
tion III we will show some of the interview data with
physics professors that lead us to see that this view of lan-
guage was really applicable to the discourse of physics. In
Section IV we will consider examples of student difficul-
ties in QM that we can explain as examples of metaphor-
ical overextension.
2. Students’ Ontological Confusion
Previously, Chi et al. have shown that many student
“misconceptions” are based on students’ incorrect onto-
logical classification of physics concepts [41, 46, 47]. For
example, physicists classify heat is a process, but students
reason with it as if it were matter.
We want to propose an extension of this idea. From our
data it appears that physicists reason about physics by
co-ordinating multiple descriptions of a particular phe-
nomenon. These descriptions may possess different or
conflicting ontological properties. For example, there
are times when physicists talk about QM phenomena
in terms of waves (a process description) whereas there
are other times when physicists prefer to talk about a
QM phenomenon in terms of particles (a matter descrip-
tion). Physicists are good at co-ordinating these different
and sometimes conflicting descriptions. Physicists under-
stand when a wave or a particle description work and how
use them appropriately in their reasoning.
Students learn these descriptions by listening to and
reading what their teachers say and write. Our sec-
ond hypothesis is that students are failing to co-ordinate
appropriately the many different descriptions that they
learn from physicists’ language. For example, physicists
often describe the “potential energy graph” in QM in
terms of physical objects (well, barrier, etc.), endowing
the graph with the properties of a physical object. Stu-
dents, hearing this language, also learn to think of the
graph as a physical object. However, students are often
unaware of when this picture is appropriate or inappro-
priate. Thus students may attach inappropriate ontolog-
ical properties to the idea of the graph as a physical ob-
ject. We hypothesize that this process leads to patterns
of student reasoning that researchers sometimes interpret
as “misconceptions”.
In Section IV we will test this hypothesis by analyz-
ing an example of students solving a QM problem and
consider several studies from the PER literature.
III. METAPHORS IN QUANTUM MECHANICS
We will trace two metaphorical systems in QM from
their origins as analogies through to modern language
that physicists use to speak and write about their
ideas. These two systems are (1) the potential well
metaphor, and (2) the Bohmian metaphor.
Both grammatical and metaphorical analyses will serve
together to illustrate a number of claims made in Sec-
tion II. The claims are: (1) Coherent systems of
metaphors exist in physicists’ language. (2) Systems of
metaphors encode historical analogies. (3) The language
encodes a representation or representations of a phys-
ical model that has an underlying ontology of matter,
processes and states. (4) Physicists use these linguistic
representations to reason productively about certain phe-
nomena.
The data for the linguistic analysis will come from two
sources. The first is the interview study with physics
professors referred to in Section II. These professors were
all native English speakers We asked them to describe
and explain various ideas in QM such as the Heisenberg
uncertainty principle, or how they would respond to a
student who asked, “what is oscillating in a QM wave?”
The interview study consisted of five subjects. The full
set of interview questions may be obtained by request
from the authors. The second source of data is a selection
of QM textbooks [19, 20, 21, 22, 23, 24].
A. The Potential Well Metaphor
1. Original Descriptive Analogy
“Because of the Pauli exclusion principle, the
electrons must be spread over the available
states; but they settle down to the states of
lowest energy, so that as more electrons are
added, the energy levels in the band fill up
like a bucket fills with water.” [48]
In this example Peierls makes the analogy explicit. The
way he uses it shows that this analogy has a descriptive
2. Analysis of Modern Language
Grammatical and ontological analysis: When
physicists speak of “potential well” and “energy level,”
they give energy an existence as water. When physicists
speak about quantum particles “leaking through a
barrier,” they give the quantum particles an existence
as water. When physicists speak of a “potential well,”
“potential step,” “potential barrier,” “confinement,”
“trap” etc. . . they give the potential energy graph an
existence as a physical object. This ontology is encoded
in the grammar. Samples of textbook writing and
physicists’ talk from interview data and accompanying
grammatical analysis are provided in Table III.
From the data we have studied, this selection of talk
and writing of physicists (Table III) is representative
of the type of language associated with the potential
well metaphor. One can see a clear pattern of gram-
mar that can be mapped to the ontological categories of
matter and processes. This is shown in Table IV below.
The grammatical analysis can tell us more than what
the objects and processes are in the metaphorical model.
It shows us that the potential well metaphor consists of
two objects, the particle/wave function, and the poten-
tial energy graph, that function as separate grammatical
participants. They interact with each other via a number
of possible processes such as “tunnel through,” “reflects,”
and so on. Thus the common grammatical structure of
the potential well metaphor contradicts the conven-
tional view of the potential energy as a property of the
particle or the system.
Metaphorical analysis The ontology encoded in the
grammar describes the basic objects and processes of the
physical model. To understand more subtle properties of
these objects, and their interactions, we need to apply a
metaphorical analysis. For this, we need to identify the
base of domains of various analogs that go into making
up the potential well metaphor.
In this section we will analyze an additional sample of
clauses and sentences from a selection of popular intro-
ductory quantum mechanics textbooks [19, 20, 21, 22,
23]. We will identify each metaphor that makes up the
metaphorical system, and present sample examples of its
use by physicists. Additional examples may be found
in [26].) Where necessary, we will identify the analogi-
cal base from where the words have been “borrowed” to
create the metaphor.
• Metaphor: The potential energy graph is a
physical object or physical/geographical
feature.
Examples:“The perfectly rigid box, represented
by a rectangular potential well with infinitely high
walls, is an ideally simple vehicle for introducing
the mathematics of quantum systems.” [21]
“Scattering from a ‘cliff’.” [23]
“. . . even for a total energy of the particle less than
the maximum height of the potential hill. . . ” [19]
“What are the classical wave analogs for particle
reflection at a potential down-step and a potential
up-step?” [21]
Base: The words “box,” “well,” “hole,” “cliff,” and
“hill” are borrowed from the category of physical
objects or physical/geographical features.
• Metaphor: The previous metaphor the poten-
tial energy graph is a physical object or
physical/geographical feature entails an-
other metaphor: The “walls” of the well or bar-
rier correspond to a physical height above the
ground. In other words, energy is a vertical
spatial dimension in the Earth’s gravitational
field. The potential energy graph is a phys-
ical object or physical/geographical fea-
ture metaphor builds on this spatial metaphor.
Examples:“It is instructive to consider the effect
on the eigenfunctions of letting the walls of the
square well become very high. . . ” [20].
Prof A: “. . . your zero point energy is going to go
up and up.”
“. . .ψ1, which carries the lowest energy, is called
the ground state. . . ” [23]
Base: The words “high,” “up,” “lowest,” and
“ground” all suggest an analogy between the verti-
cal axis of the potential energy graph and a vertical
spatial dimension on the Earth’s surface.
• Metaphor: The potential energy graph is
a container. The potential energy graph “con-
tains” or “traps” either the wave function, the par-
ticle or the energy of the particle.
TABLE III: Samples of physicists’ speech and writing for grammatical analysis.
Sample of physicist’s speech or writing Simplified exerpt with analysis
“In both cases, a classical particle of total energy E. . .moves
back and forth between the boundaries.” [21]
a classical particle [medium] moves back and forth [pro-
cess:event] between the boundaries [circumstance:location].
“. . . when you have a confined system, . . . [the width of the
box is] going to set the scale for what . . . the magnitude of
the energy is, so as you confine it [the particle] more and
more, your zero point energy is going to go up and up.” -
Prof. A, interview study
. . . your zero point energy [medium] is going to go up and up
[process:event].
“. . . it has been seen that potential barriers can reflect parti-
cles that have sufficient energy to ensure transmission clas-
sically.” [19]
. . . potential barriers [agent] can reflect [process:event] parti-
cles [medium]. . .
“This [wave] packet would move classically, being reflected
at the wall. . . ” [22]
This wave packet [medium] is reflected [process:event] at the
wall [circumstance:location].
“The α-particle then ‘tunnels through’ the barrier. . . ” [19] The α-particle [medium] then ‘tunnels through’ [process] the
barrier. . . [range]
“. . . they [α particles] start out with the energy E inside the
nucleus and ‘leak’ through the potential barrier.”[24]
α particles [medium] leak through [process:event] the poten-
tial barrier [range].
“. . . the phenomenon of tunneling. . . allows the particle to
‘leak’ through any finite potential barrier. . . ” [23]
The particle [medium] leaks through [process] any finite po-
tential barrier [range].
TABLE IV: Ontology of the potential well metaphor
Matter Process
QM/classical particles, wave packet,
energy, energy walls, energy barrier,
potential barrier, barrier
moves, reflect(s),
tunnels through,
leaks through
Examples: “The exponential decrease of the wave
function outside the square well for the second
energy state is less rapid than is the correspond-
ing decrease for the lowest energy state as indi-
cated. . . ” [21]
“Inside the well where V (x) = −V0. . . ” [23]
“. . .bound [energy] states in the. . . well” [21]
Base: Words such as “well,” “confined,” “bottle,”
and “bound” all suggest an analogy to some sort
of container. Physicists also make a distinction
between “in/inside” and “outside” the well: Such
adverbial phrases also indicate the presence of the
container metaphor.
Various other elaborated metaphors are built on this ba-
sic set. Examples of their usage may be found in [26].
• The potential energy graph is a barrier.
• The potential energy graph is a hard
container/barrier or the potential energy
graph is a semi-hard container/barrier
• The particle, the wave packet, and the energy are
all given an ontological status of matter. More
specifically:
– QM particles are hard objects.
– The wave packet is a soft or breakable
object
– QM particles are a fluid
– The energy is a fluid
See Table V for a summary of the metaphorical map-
ping from the domain of physical/geographical features
to the domain of quantum systems that involve an inter-
action between two or more objects.
3. Productive Modes
How do physicists piece together the gram-
mar/ontology of the potential well metaphor?
How do they use the associated imagery to reason
productively about quantum systems? From the dis-
course of professors and textbooks we have identified the
presence of productive modes for the potential well
metaphor. We present five examples below:
Squeezing: Squeezing the walls of the well forces the
water upwards, thereby raising and spacing out the “en-
ergy levels.”
Example: Prof. A: “. . . when you have a confined sys-
tem, . . . [the width of the well is] going to set the scale for
TABLE V: Summary of the metaphorical mapping between
the base domain of physical/geographical features and the tar-
get domain of interacting QM systems
Base domain Target domain
Physical/geographical
features
Interacting QM systems
Physical or geographical
features
→ Potential energy graph
Vertical height of physi-
cal/geographical feature
→ Magnitude of energy at a
point/region on the poten-
tial energy graph.
Hardness or softness of a
→ “Height” of the potential en-
ergy graph
Container with top face
→ “Trapping” of QM particles,
“bound” states
Billiard ball → QM particle in some
circumstances
Soft or breakable objects → QM wave function or wave
packet
Fluid → QM particle in some circum-
stances, or the energy of the
particle/system
Ball bounding off a wall → Reflection of QM particle
Tunneling/penetration → Process by which a QM
particle “passes through” a
seemingly solid “barrier”
Leaking → Process by which a QM par-
ticle “escapes” from a QM
“container”
what . . . the magnitude of the energy is, so as you confine
it more and more, your zero point energy is going to go
up and up.”
Stacking: Matter takes up space. Filling up the
well/bucket can be used to understand the behavior of
fermions.
Example: We already observed Peierls make this anal-
ogy explicit[48]. The following is an example of Prof. A.
using it: “if you have fermions then. . . you have to keep
stacking the fermions into levels which get more and more
elevated in energy. . . ”
Tunneling/leaking: The potential energy graph be-
haves as a physical container or barrier, preventing the
escape of the particle. This leads to the ideas of “tunnel-
ing” or “leaking.” When reasoning productively, physi-
cists recognize that a higher or wider barrier means less
probability of tunneling.
Examples: Feynman et al. write: “. . . they [α-particles]
start out with the energy E inside the nucleus and ‘leak’
through the potential barrier.”[24]
Griffiths writes: “If the barrier is very high and/or
very wide (which is to say, if the probability of tunneling
is very small), then the coefficient of the exponentially
increasing term (C) must be small. . . ” [23].
Reflecting/scattering: The wave function or particle
is reflected by or scatters off a hard barrier.
Example: “For this reason, the rectangular potential
barrier simulates, albeit schematically, the scattering of
a free particle from any potential.” [22]
A way of speaking: It is difficult to come up with
realistic physical systems of quantum mechanical parti-
cles without resorting to lengthly descriptions. (See [21]
for examples.) By separating the QM particle from its
potential energy graph, physicists are able to talk easily
about the particle interacting with an external object (its
own potential energy graph).
Example: During one of the interviews we asked a pro-
fessor to describe the process of trapping and cooling
atoms to absolute zero.
DTB: “Are the atoms going to jump out, are
you not going to be able to trap them?”
Prof. E: “No, of course not, you’d just go
down to the lowest eigenstate. I mean, I don’t
know how they were trapped in the first place,
but suppose you had them in a square well for
example.”
4. Summary
We have tried to illustrate how grammar and metaphor
work together to encode the features of a particular de-
scriptive model. Each aspect is necessary and the gram-
matical and metaphorical analysis together serve to illu-
minate features that each individual analysis cannot do
on its own.
Fig. 3 presents a visual summary of how the po-
tential well metaphor is structured. The poten-
tial well metaphor is an example of a metaphor-
ical system made up of three ontological metaphors,
two of which are encoded in the grammar (the po-
tential energy graph is a physical object or
physical/geographical feature, and the parti-
cle/wave function/energy is a physical object
or matter), and one which can only be identified by
looking at the imagery (energy is a vertical spatial
dimension). Other metaphors such as the potential
energy graph is a container or the potential en-
ergy graph is a hard barrier build on and elab-
orate this ontology. At the sentence level, we can see
how productive modes of reasoning are formed by intro-
ducing grammatical processes through which the particle
or wave function interacts with its own potential energy
graph. These productive modes are squeezing, stacking
tunneling/leaking, and reflecting/scattering. Physicists
also use the potential energy graph is a physical
object metaphor as a substitute term (or metonym) for
the actual physical QM system.
FIG. 3: Summary of the metaphorical system and its usage by physicists.
B. The Bohmian Metaphor
1. Introduction
The potential well metaphor, as a linguistic repre-
sentation, has many of the characteristics of a physical
model as described by Etkina et al. [44]. The language
describes objects with properties, and processes by which
those objects interact with each other. In contrast, the
Bohmian metaphor has almost none of those character-
istics. It seems to exist in the language of physics solely
as a way of speaking. It is an interesting case because
it is easy to identify the metaphor, but not the original
analogy. Therefore, in this section, we will present the
linguistic analysis first and the study of the analogy on
which it is based, second. Although we have called it
the “Bohmian” metaphor in honor of David Bohm, who
advocated the Bohmian interpretation of QM, the entry
into the language of physics can be traced back much
earlier than this.
2. Modern Language
The Bohmian metaphor is identified in language by
words and phrases that suggest that the wave function or
quantum state is a container that contains the quantum
mechanical particle. There are only two metaphors that
make up the Bohmian metaphor:
• Metaphor: The wave function/quantum
state is a container.
Examples: Noun groups such as “wave packet”
or “envelope function” indicate an analogy to a
container.
• Metaphor: The QM particle is a physi-
cal object contained inside the wave func-
tion/quantum state.
Examples: This is suggested by prepositional
phrases such as “in the ground state” in sentences
such as “The electron is in the ground state.”
Connection to grammar: In the Bohmian
metaphor the wave function or quantum state is
conceived of as a container that has a particle as a
separate entity inside it. The language is based on two
sources. The first source is an analogy to Einstein’s
ghost field idea (see Section III B 3 below) but the second
source is language itself. Cognitive linguists hypothe-
size that mental states are spoken about in language
metaphorically as containers [25]. For example, if one
is depressed one can say, “I am [relational process] in
a state of depression [location].” Such statements seem
to all have the same grammatical structure, namely a
relational process followed by circumstance of location. It
seems as if ontological physical states are expressed by an
identical grammatical structure: such as, “the electron
is [relational process] in the ground state [location].”
It seems as if physicists have unconsciously borrowed
this grammar that expresses mental states in every day
experience and used it to express physical states in
physics. As mentioned in Section II, the metalingual
apparatus that we have for realizing physical states in
language appears to be extremely limited. The states
are locations metaphor, supported by this unique
grammatical structure, is one of these limited means
of expression. A statement about the physical location
of an object within another object would be classified
in the ontological category of matter if taken literally.
Metaphorically a statement such as “the electron is in
the ground state,” is a statement about the energy of
a quantum system, and physicists recognize energy as
a state function. There is a clear ontological conflict
between the literal interpretation of the statement
and the meaning that is intended. This leads us to
hypothesize that such statements will cause students
confusion and may lead to difficulties.
3. The Original Analogy
In the case of the potential well metaphor, the
analogy on which it is based, is relatively well under-
stood. The Bohmian metaphor is easy to identify, but
the original analogy is not well known. If our frame-
work is correct and language is built on analogy then an
original analogy should exist in the mainstream QM lit-
erature. We started searching the original QM papers in
the hope that we would find some explicit reference to
the idea that the wave function could contain the parti-
cle inside it. Remarkably, we found such a reference in a
paper by Max Born, published in 1926 [16].
“Neither of these two views seem satisfactory
to me. [Heisenberg’s interpretation of the
wave function and the Schrödinger/deBroglie
interpretation of the wave function] I would
like to attempt here a third interpretation and
test its applicability to collision processes. I
thereby pin my hopes on a comment of Ein-
stein’s regarding the relationship between the
wave field and light quanta. He says roughly
that the waves may only be seen as guid-
ing [showing] the way for corpuscular light
quanta, and he spoke in the same sense of a
“ghost field.” This determines the probability
that one light quantum, which is the carrier of
energy and momentum, chooses a particular
[definite] path. The field itself, however, does
not have energy or momentum.” [16] [Trans-
lation by D.T.B.]
There are several remarkable features about this pas-
sage from Born:
• Firstly, it lays out the Bohmian interpretation of
quantum mechanics twenty-five years or more be-
fore Bohm proposed the same idea, and one year
before deBroglie’s attempt at a “pilot wave” the-
• Secondly, when Born says “Neither of these two
views seem satisfactory to me,” he is referring
to (1) the Heisenberg interpretation of QM which
Born describes as “an exact description of the pro-
cesses in space and time are principally impossi-
ble,” and (2) the Schrödinger/deBroglie interpreta-
tion which Born summarizes: “He tries to construct
wave groups which have relatively small dimensions
in all directions and should, as it seems, directly
represent moving corpuscles.” Born is cautioning
against overly literal interpretations of (1) an anal-
ogy to a classical particle (Heisenberg’s approach),
or (2) an analogy to a physical wave (Schrödinger’s
approach). Born suggests that both views lead to
untenable positions in the physical interpretation of
QM and introduces a third model which is essen-
tially a hybrid of the wave and particle analogies.
Born makes, an analogy to Einstein’s interpreta-
tion of light waves and light quanta and applies it
to particles with non-zero mass.
Born’s mode of reasoning appears to be metaphor-
ical as well as analogical. He makes an analogy
to Einstein’s view of the electromagnetic field as a
ghost field, but he does not suggest that the wave
function is “like a guiding field.” Rather, he ex-
presses Einstein’s idea directly as “. . . the waves
may only be seen as guiding the way for corpus-
cular light quanta. . . ” [our emphasis]. For Born to
interpret the wave function as a probability distri-
bution, he felt it necessary to blend together a wave
picture and a particle picture with real particles
who have definite trajectories determined proba-
bilistically by the wave function. Lakoff and Núñez
refer to such a mental construct as a metaphorical
blend [28] after the conceptual blend of Fauconnier
and Turner [31].
• Thirdly, Born is aware of the limitations of the
metaphorical picture he has introduced. In blend-
ing a wave and particle picture into a model that
looks and feels like a statistical ensemble, Born cau-
tions about taking this “Bohmian” picture too lit-
erally when he writes: “However, the proposed the-
ory is not in accordance with the consequences of
the causal determinism of single events.” [16]
4. Productive Modes
One of the difficulties with QM is the question of how
to speak about quantum processes meaningfully. We sug-
gest that the Bohmian metaphor permits a partial so-
lution to this problem. Although Born’s suggestion (in-
tepreting the wave function as a pilot wave) never made
it to the mainstream of physics, the associated language
is now ubiquitous and used productively by physicists as
we will show in the following example:
D.T.B.: “. . . if you wanted to think about how
an electron propagates. . . It wouldn’t be sen-
sible to talk about it as a wave, you would
think more as a particle?”
Prof C: “. . . you can think of it as a plane
wave. Yeah, . . . in an envelope function
which makes it into a wave packet.”
More examples may be found in [26].
IV. STUDENT DIFFICULTIES
A. The Potential Well Metaphor
A group of four junior students in their first QM course,
were video taped while working on their QM homework
problems. All students were native English speakers.
The discussion we present is centered around a problem
from French and Taylor[21]. The question was: “What
are the classical wave analogs for particle reflection at
a potential down-step and a potential up-step?” Notice
here the potential well metaphorical system serving
a specific function: namely, it describes the shape of the
potential energy graph (“potential down-step”).
S1: Well, there wouldn’t be reflection in par-
ticle physics on a down-step right? Or even,
I don’t think even on an up-step. . .
S3: No, there’s reflection on an up-step, total
reflection.
S1: Not classical though, right?
S2: Not if its less than the energy though.
S1: It just slows it down.
In this opening exchange we can observe S1 talking at
cross purposes with S2 and S3. S2 and S3 seem to be
imagining a classical particle approaching the step and
bouncing back (later dialogue show that they do not re-
ally shift from this literal view of the situation), while S1
seems to be thinking of a wave approaching with energy
greater than the energy of the step. As we see later, S1
is reasoning from picture of a surface water wave passing
over a step in a river or sea bed.
S1: Not quite sure what the wave analogs
would be. If I had to guess I’d say it would
be like sound, like those things that male
cheerleaders have, like big cones.
S4: Megaphones?
S1: Yeah. ’Cause I think, you
know,. . . basically a step up or step down in
resistance. But I am not quite sure what we
are supposed to say about that.
This is the first example of an analog from S1. It is
interesting that S1 sees the key as a change in resistance
(at the end of the first exchange S1 says “It just slows
it down”), yet he still is the one who proposes a phys-
ical form (consistent with the ontology of the graph as
a physical object) surrounding the medium rather than
a change in the medium itself (which would represent a
more obvious change in resistance for the wave).
S2: So they’re saying that there would be re-
flection on a potential up-step like a. . .
S1: Yeah, just like a sound, or a water wave
or something.
S1: Um, well ’cause I know on a potential up-
step,. . . like if you just had. . . water and you
had, you know, deeper part and a shallower
part, and you had a wave, some of it would
reflect back.
Here S1 applied the metaphor of a physical object
again, and proposes a second analog based on the physi-
cal form of the graph rather than a change in “density”
or “tension” of the medium. Actually, a physical step on
a river bed could be a valid example if S1 connected it to
a model of how the resistance experienced by a surface
wave attenuates with the depth of the water. He does
not, and this explains his uncertainty below.
S1: So that’s not too hard to see. But like, I
would guess that the same thing would hap-
pen if you had a down-step, but that’s not
something like I really, I could vouch for. Like
I think they’re looking for stuff that like most
people know.
S2: Is that what its saying? Its coming at
it with every energy, like continuous energies,
like around the step?
S2’s statement is interesting. The use of “at” and
“around” are examples of grammatical location and sug-
gest the metaphor: the step is a physical object.
S1 shows he is still on the right track when he says:
S1: I think they’re just asking for like,
examples from. . . in real life from when a
wave. . . goes into a space of less resistance and
has reflection back.
S4: So in classical what would happen at a
potential down-step?
S1: A potential down-step?
S2: It would just keep going. . .
S1: . . . It would just speed up. At a potential
up-step it would just slow down.
1. Discussion
One alternative hypothesis to explain the difficulties
presented above could be that the students are unable
to interpret the physical meaning of the potential en-
ergy graph or are simply not understanding the situation.
However, S1’s ability to interpret potential energy graphs
correctly and articulate the key to the analogy discounts
this hypothesis. The data show that his inability to come
up with a productive analog must be based on other fac-
tors. Our framework explains how S1 is distracted by ap-
plying an overly literal interpretation of the potential
well metaphor in an inappropriate situation. Possibly,
a way of talking (i.e., describing the potential graph as
a “step”) is affecting students’ reasoning. Our analysis
(Section IV above) shows that the students in this group
are searching in the category of “physical objects” for an
analogy, in accordance with the underlying ontological
metaphor the potential energy graph is a phys-
ical object rather than searching in a more produc-
tive category. Other researchers have also noticed that
QM students tend to pick 2-d gravitational analogs when
asked to come up with physical examples of 1-d potential
energy graphs [49, 50].
As a control we posed the same problem to the profes-
sors in the interview study. They all responded that an
analogy of an electron beam scattering off of a potential
down step is light traveling from a medium with greater
index of refraction to a medium with a lesser index of
refraction. When asked why changing optical media was
a good analog, most were unable to explain, but contin-
ued to elaborate their answer. Only one professor was
able to explain why this was a good analogy. Prof. E:
“I know because we’ve thought about these things before
and its just been classified in that category.” This state-
ment suggests that physicists are able to automatically
search for an analog in a category of analogous processes
rather than analogous objects. It may also suggest that
physicists’ ideas have become so tightly bound into larger
conceptual units that professors are unable to break down
their reasoning into smaller parts again.
We have shown how physics professors can use
metaphorical systems to reason productively in certain
situations while students take the same representation
and apply it too literally and inappropriately in other
situations. Strange ideas like the megaphone make sense
if we understand the underlying ontology of the graph,
spoken of as a physical object. We think that the ex-
ample of student discourse presented above is a typical
example of students’ difficulties arising from linguistic
representations.
2. “Robust Misconceptions” Related to the Potential
Well Metaphor
Are there “robust misconceptions” in QM? The char-
acteristics of a robust misconception are that it must be
(a) present before instruction, (b) common to a signifi-
cant percentage of students in a particular class, and re-
producible in form and structure across different classes
at different institutions in different contexts, and (c) re-
sistant to instruction. (See [51] for example.)
Although research on students’ understanding of QM
is in its infancy, it appears that students do have spe-
cific difficulties that have the characteristics of a robust
misconception. One emerging example is presented in
Table VI. It has been observed that students think that
a QM particle loses energy when it tunnels through a bar-
rier. McKagan et al., who studied this example, freely
use the word “misconception” in their paper [52].
A consistent pattern of reasoning is presented in Ta-
ble VI. This pattern contains the following two elements:
(1) It takes energy for a particle to tunnel through a
barrier. (2) Making the barrier wider or higher means
that the particle loses more energy/expends more effort
when tunneling through it. Morgan et al. speculate that
the difficulty may come from either (a) intuitive classical
ideas about a particle passing though a barrier, or (b)
physicists tend to draw the potential energy graph and
the wave function superimposed. Thus a decaying wave-
function amplitude may be confused with a decrease in
energy.
McKagan et al., however, noticed something interest-
ing in their study. In interviews, they discovered that
students do not see the potential energy graph as repre-
senting the potential energy of the particle in question.
They see it rather as some external object with which
the particle interacts. The authors describe an example
from their interviews:
“When pressed, he said that the ‘bump’ was
‘the external energy that the electron inter-
acts with’ and insisted that it was not the po-
tential energy of the electron itself, in spite of
the fact that it was explicitly labeled as such
in the previous question.”
The authors speculate that statements like “a particle in
a potential” may be the cause of this problem.
Our analysis supports this idea and provides an expla-
nation for the underlying causes of this student difficulty.
The problem is much more widespread than just phrases
like “a particle in a potential.” As we pointed out in
Section III A 2, many statements that fall under the cate-
gory of the potential well metaphor, tend to separate
the particle or wave function from its potential energy
graph in the grammar of the sentence. Most often the
particle/wave function functions grammatically as the
medium while the potential energy “barrier” functions
as either the range, or circumstance of location. The two
grammatical participants then interact with each other
by a grammatical process such as “tunnels through” or
“is reflected.” We hypothesize that the language is the
primary source of the students’ model. Graphical repre-
sentations (such as the superposition of the energy graph
and the wave-function) and classical intuitions build on
and extend this basic model, leading to the idea that
energy is lost in the tunneling process.
TABLE VI: Selected examples of the “exhaustion” misconception: Summary from three studies.
Authors’ summary and explanation Sample student responses used to justify this
explanation.
Lei Bao [49] interviewed ten students over two semesters.
Three responded with the incorrect idea that a quantum
mechanical particle loses energy when it tunnels through a
potential barrier.
Bao observed that all three students gave similar re-
sponses. Mike: “. . . less energy so the amplitude will be
reduced,. . . Amplitude is reduced because energy is lost in
the passage [our emphasis]. . . ”
Jeffrey Morgan et al. [53] found that all six students that
they interviewed thought that the particle lost energy when
it went through a potential barrier. Two of the students had
completed a senior level QM course and four had completed
a sophomore level introductory QM course
Selena: “Uh, because it requires energy to go through this
barrier.”
Jack: “. . . when the particle of some . . . energy, encounters a
potential barrier, there is a possibility. . . that a particle will
actually just go straight on through, losing energy as it does
so, and come out on the other side. . . at a lower energy. . . ”
McKagan et al. [52] gave a conceptual test to a group of
engineering majors (N = 68) and physics majors (N = 64)
after they had completed a modern physics course. One of
the questions probed students’ understanding of tunneling
processes. On this question 24% of the engineering majors
were able to answer correctly and 38% of the physics majors
were able to answer correctly.
No interview samples were provided, but the authors sum-
marize the student responses as follows: “all students who
selected answers A, B, or E [more than 50% for both en-
gineers and physicists] argued that since energy was lost in
tunneling, making the barrier wider and/or higher would
lead to greater energy loss.”
3. Summary
The example of the potential well metaphor illus-
trates how the language used to describe certain QM sys-
tems may pose extraordinary difficulties, especially if stu-
dents are not aware of how and why metaphorical terms
are being used. The metaphorical language, grounded in
the classical world, may encourage students to associate
extra (classical) properties to the QM system as they try
to coordinate these new representations with their prior
understanding of the world. These over-extensions of the
representation seem to be the source of their difficulties.
B. The Bohmian Metaphor
As part of our study, two senior undergraduate physics
majors (in their second QM course) agreed to be video-
taped while working on their QM homework together.
Both were native English speakers. In this particular ses-
sion S1 and S2 were working on a problem worked out in
class by the lecturer that they did not understand. The
question may be expressed as follows: “Given an electron
in the ground state of an infinite square well of width L.
The walls are suddenly moved apart so that the width
of the well becomes 2L. What is the probability that the
electron is in the ground state of the new system?”
The two students working on the problem understood
the sudden approximation, they calculated the overlap
integral and got a numerical answer which was reason-
able. Then S1 stopped and pondered that his answer
made no sense. He argued that his answer should be
zero. A discussion with the observer (D.T.B.) followed.
S1: But I am still confused about what I
was. . . saying about if there is a probability
FIG. 4: Wave function of the electron in the sudden approx-
imation
that it is in the [sic] first ground state — it
seems to say that the particle can be where
it is not.
D.T.B.: Why do you say that?
S1: Because we know that the wave function
looks like this [points to a sketch similar to
Fig. 4] — Oh, so its not the probability of it
being in the ground state really. . . I think the
probability is really. . . I mean, we know that
its in this state [points to sketch similar to
Fig. 4] so it can’t be in the ground state. So
it’s zero [the probability].
The discussion circled around this theme for some
time. S1 was concerned that if the particle was “in the
ground state” of the new well, it would permit the parti-
cle to exist outside of the [-L/2,L/2] region of its initial
wave function. The wave function limits where the par-
ticle can be, but to say the electron is “in the ground
state of the new well” does not suddenly permit it to ex-
ist outside of the [-L/2,L/2] region; it is simply a state-
ment about measuring the energy of the electron. We
believe that the linguistic framework we have developed
provides both a reasonable and parsimonious explana-
tion for S1’s difficulties. The prepositional phrase,“in the
ground state,” is functioning grammatically as a location.
S1’s argument, that the probability should be zero, draws
specifically on the location metaphor. He says, “it [the
original question] seems to say that the particle can be
where it is not.” This statement suggests that he is view-
ing the question as a question about the location of the
particle. In other words, he is interpreting the phrase “in
the ground state” literally rather than figuratively.
This difficulty with the Bohmian metaphor remains
undocumented in the physics education research liter-
ature. However, a physics professor who teaches un-
dergraduate quantum mechanics, reported in a private
conversation that he observed the identical difficulty
amongst his students with the same sudden approxima-
tion problem.
V. CONCLUSION
We have shown that coherent systems of metaphors
exist in physicists’ language. We have shown that physi-
cists use these metaphorical systems in their language
to speak and reason productively about QM systems.
They are able to invoke many different metaphorical sys-
tems, sometimes with apparently conflicting ontologies,
depending on the situation they are trying to describe.
At the same time, physicists appear to understand the
applicability and limitations of their metaphorical lan-
guage in each situation. We have also shown how these
metaphorical systems can be identified with systematic
use of both grammatical and metaphorical analysis. And
we have shown how the elaborated metaphors build on
the underlying ontology encoded in the grammar.
In some cases, it seems that physicists have appro-
priated conceptual metaphors from language to express
their ideas. The example with Born and the Bohmian
metaphor shows how a new idea in physics comes out of
a blending of older ideas into a metaphorical blend. Like-
wise the final product of the language is (in this case) a
blend between an analogy to Einstein’s ghost field and
also already existing structures in language that are nor-
mally used to describe ontological mental states.
We have presented two case studies of groups of stu-
dents struggling with and being confused by overly lit-
eral interpretations of the metaphorical language they
encounter in QM. The context of QM is particularly con-
vincing because it is difficult to argue that students enter
their QM course with preconceptions or misconceptions
about QM based on personal experience. Many of the
difficulties observed, appear after instruction. It seems
more plausible to hypothesize that these difficulties are
related to the way in which physical ideas are presented
during instruction itself.
We have presented one example (the exhaustion mis-
conception) of a documented common conceptual diffi-
culty that students have with QM and how we can ac-
count for their näıve model with the linguistic framework
we have developed. Physicists understand that higher
barrier means a slower rate of tunneling or leaking. In
contrast, students think that the particles get tired. The
underlying issue is use and misuse of the metaphorical
picture.
VI. FUTURE DIRECTIONS
We feel that further research on the role of language in
learning physics needs to examine more carefully the in-
structional implications of language as a legitimate repre-
sentation of knowledge and ideas in physics. For example,
how can we make students more aware of the presence
of physical models encoded in the metaphorical language
that we use? Can students be encouraged to think about
the applicability and limitations of different metaphor-
ical pictures [40]? Instead of allowing students to say,
“the electron is trapped in a square well,” unchallenged,
maybe the most important question to ask students is,
“what do you mean, what is this ‘square well’ you are
talking about?” As a corollary, maybe we should en-
courage students to ask us, “what do you mean?” when
we use a metaphor such as the electron is a wave
without justifying why it is applicable and when it is not.
Does it matter how we ask questions of our students?
If we phrase a question with different grammar or differ-
ent metaphors, do students respond differently? There
maybe occasions when the way in which the question is
asked is obscuring the real physical understanding that
students have. There is some preliminary evidence that
this may indeed be the case [54].
It seems to us that if we think of language as a repre-
sentation and recognize its unique difficulties, we should
put more effort into helping students become comfort-
able with this representation. Future research could fo-
cus on student difficulties in other areas of physics that
may be related to the language that students hear in the
physics classroom. (See [13] for example.) In some cases
difficulties may be related to linguistic models that stu-
dents have developed prior to instruction. We suggest
that some student difficulties may be attempts to nego-
tiate the meaning and applicability of different linguistic
models. Awareness of such linguistic difficulties would
help teachers to facilitate their students’ learning. More
research on this idea is needed.
There is one major aspect of cognitive linguistics that
we have not attempted to apply to the field physics edu-
cation research in this paper. This is the idea of concep-
tual blending [31]. Conceptual blending may provide a
complementary account of many of the ideas in this pa-
per. Conceptual blending also has an added advantage
in that it could account for online meaning construction
in terms of the blending of metaphors. This may bet-
ter account for “local” or “personal” ways of expression
observed among individual professors and students. The
dynamics of blending may also be useful for answering
questions about how we can make students more aware
of the myriad of models encoded by the metaphors in
physicists’ language. We think that this may be a fruit-
ful line of inquiry in future work.
Acknowledgments
We would like to thank the following people for their
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  This paper introduces a theory about the role of language in learning
physics. The theory is developed in the context of physics students' and
physicists' talking and writing about the subject of quantum mechanics. We
found that physicists' language encodes different varieties of analogical
models through the use of grammar and conceptual metaphor. We hypothesize that
students categorize concepts into ontological categories based on the
grammatical structure of physicists' language. We also hypothesize that
students over-extend and misapply conceptual metaphors in physicists' speech
and writing. Using our theory, we will show how, in some cases, we can explain
student difficulties in quantum mechanics as difficulties with language.

Introduction
Our theory was developed from a number of sources of
data: (1) Interviews with 5 physics professors. (2) Orig-
inal QM papers from Born [16] and Schrödinger [17], as
well as an analysis from Goldstein [18] of how Schrödinger
developed the wave equation. (3) A selection of older and
more modern, popular introductory quantum mechanics
textbooks [19, 20, 21, 22, 23, 24]. (4) Two physics stu-
dent homework study groups.
We began by comparing the way Schödinger and Born
wrote about their ideas with the way modern textbooks
and physics professors wrote and spoke about the same
ideas. This led us to define two separate patterns of
language used to express ideas in QM:
1. The first pattern was language used by the in-
ventors of QM. They tended to use cautious and
figurative language. Ideas were often expressed
as comparisons of the form “X is like Y in cer-
tain respects.” They made analogies explicit and
cautioned against overextending or misinterpreting
these analogies.
2. The second pattern we observed was language used
to communicate already established knowledge of
QM. (Language used by physics professors and
modern QM textbooks.) This language was char-
acterized by statements of fact with little if any ref-
erence to the original analogies on which the ideas
were based.
These two patterns of language lead us to investigate the
role of analogy and metaphor in describing physicists’
language.
In our data of students working on their QM home-
work problems, we focussed our attention on episodes
when students stopped calculating, and engaged in an
activity that could loosely be described as sense-making.
In these episodes it appeared to us as if students under-
stood the physical ideas, but they were confused about
the language used to express the physical ideas. We hy-
pothesized that students were confused by the figurative
language that physicists used to describe their ideas. We
will justify this claim in Section IV.
B. Analogical Models Encoded as Metaphors in
Physics
1. Metaphors in Physics Language
Lakoff and Johnson [25] have hypothesized that hu-
man language and the human conceptual system are
largely made up of unconscious conceptual metaphors.
We have extended this idea to physics by suggesting that
physicists speak and write using conceptual metaphors.
For example, physicists talk about “diffraction of elec-
trons” and a “wave equation for the electron.” Both
phrases suggest the conceptual metaphor the electron
is a wave. Conceptual metaphors are often unconscious
metaphors and seldom made explicit. They have become
quite literal, losing their figurative origin through their
unconscious and frequent use. For a more complete dis-
cussion of what a metaphor is and how it differs from
analogy and simile, we refer the interested reader to [26].
Other excellent discussions of the theoretical status of,
and issues surrounding, metaphor and analogy may be
found in [27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
2. Types of Analogies Encoded as Metaphors
Researchers have shown that physicists use analogical
models to construct new ideas [37, 38]. These analogical
models become, in time, encoded linguistically as concep-
tual metaphors [39, 40]. The way physicists talk about
already established knowledge is different than the way
they talk about new ideas they are trying to comprehend
themselves.
We will take this idea further. From the primary data
(textbooks, original papers, and interviews with physics
professors), we have identified three types of analogical
model that metaphors encode. These can be classified by
their origin and function:
1. Current analogical models: For example,
Schrödinger based his wave equation on an anal-
ogy to wave optics [17, 18]. The corresponding
metaphorical system is the electron is a wave
and is spoken about by modern physicists in terms
such as “electron interference,” “electron diffrac-
tion,” “wave equation,” and so on.
2. Defunct analogical models: It is often the case
in physics that older models, whose limitations
have been experimentally exposed and supplanted
by better models, live on in the language of physics.
The caloric theory of heat lives on in phrases which
reflect the heat is a fluid metaphor. For exam-
ple, “heat flows from object A to object B.” Physi-
cists use these metaphorical pictures when they rea-
son. We will elaborate this point further below.
3. Descriptive analogies: For example an anal-
ogy between a physical valley and a potential en-
ergy graph. The metaphor is potential energy
graphs are water wells. Examples of how the
metaphor is used in language are: “potential well,”
“potential step,” “energy level,” “ground state,”
and so on.
We will identify metaphors that encode analogies 1 to
3 by identifying the base of the analogy [35]. We will use
the idea that conceptual metaphors borrow terms from
the base of the analogy and apply these words directly to
the target concept. For example, if we look at the matter-
wave analogy in QM, we can consider that a water wave
or an electro-magnetic wave is the prototypical example
which will serve as the “base” of the analogy. Thus words
such as “interfere,” “polarize,” “diffract,” and “wave” are
used in the context of “an electron.” Such examples will
be identified as instances of the electron is a wave
metaphor.
3. Features and Functions of Metaphors in Physics
We hypothesize that physicists unconsciously prefer to
speak and write in metaphors because metaphors have
certain features and functions that are advantageous to
them. The features and functions of these metaphorical
systems are listed below with examples from interview
data with physics professors.
Feature 1: Conceptual metaphors encode analogies.
They encode a more deep and complex piece of knowledge
which is the completely elaborated analogy. That elabo-
ration as an analogical model is, however, tacit amongst
the community who use the metaphor and associated
model regularly.
Function: Physicists are able to use these metaphor-
ical systems to reason productively about a particular
situation or problem. For example, the electron is a
wave metaphor can be used productively to explain the
Heisenberg uncertainty principle:
“I often think of it. . . in terms of Fourier
transforms and the reciprocity between the
bandwidth of the channel and the length of
the signal pulse that can be detected.” (Prof
Note the use of words from the base domain of electro-
magnetic waves: “Fourier transforms,” “bandwith,” and
“signal pulse” in particular.
Even defunct analogies (type 2) represent productive
modes of thought for physicists. There is a class of prob-
lems for which it is quite adequate to talk about heat
as a fluid. For example, when there is no work being
done on or by the thermodynamic system, it is satisfac-
tory to think of heat flowing into or out of the system and
that the change in temperature of the system is directly
proportional to the amount of heat gained or lost.
Feature 2: Metaphorical systems are partial in na-
ture. This means that more than one metaphorical sys-
tem is needed to fully understand a physical concept.
Function: We observed that physicists switch easily
and unconsciously between one system and another de-
pending on the type of question that is asked. For exam-
ple, in the following extract, Prof. D switches back and
forth between particle and wave metaphor to describe the
process of electrons passing through a Young’s double slit
apparatus.
“Of course in any one experiment,. . . you will
not observe. . . an interference pattern on the
screen [wave metaphor] — if all you do is to
scatter one electron [particle metaphor]. The
intensities are just too low [wave metaphor].
. . . you have to have a large number of elec-
trons [particle metaphor], you have to have a
beam of electrons [wave metaphor]. And each
electron will contribute a little piece of the in-
tensity that you see on that screen [particle
metaphor]. What I envisage is. . . a beam of
electrons which can be represented by a plane
wave [wave metaphor]. . . ”
Feature 3: Metaphors involve the use of the verb “is”
rather than “is like.” Metaphors are grammatically iden-
tifying relational processes, i.e., they are grammatically
equivalent to statements of category membership.
Function: We hypothesize that metaphor reflects a par-
ticular aspect of an expert physicist’s thought process.
The use of metaphor itself rather than simile is signifi-
cant. Irrespective of deep philosophical discussions about
what is “real,” it seems apparent that physicists them-
selves need to assert something stronger than “like” —
they need to assert “is” in their own reasoning process.
We suggest that this is a fundamental trait of how knowl-
edge is generated and represented in physics. It is signifi-
cant because such assertions may often conceal the vague
or partial nature of metaphor itself.
For example, Prof. D provided the following response
to the question: What happens to a single electron when
it passes through a Young’s double slit apparatus?
“. . . [to] understand that experiment, you’ve
got to forget about the idea that an electron is
a particle. It is not a particle in that context,
it behaves like a wave. So you just think of
it as a plane wave [our emphasis] advancing
on the two slits, and the interference between
the two. . . outgoing beams, just using Huy-
gen’s principle, leads to the. . . interference
pattern. . . ”
Note that comparison, “it behaves like a wave,” is fol-
lowed directly by, “just think of it as a plane wave.”
Feature 4: The apparatus of language constrains the
ways physicists can talk about physical phenomena and
therefore constrains the types of models that can be rep-
resented in language.
Function: Descriptive analogies (type 3) encoded
as metaphors also represent ways of speaking
about/describing physical systems. This is very
important because there is a limit on what can be
represented with language. Such metaphors also give
abstract concepts and quantities a grounding in physical
reality and physical experience.
Consider for example, the modern physicist’s view of
energy. Physicists can define energy as a state function
yet can physicists speak literally about energy as a state
function? Our hypothesis is that it is simply impossible
to come up with grammatical constructions that convey
the meaning of energy as a state function. The very best
locutions are “energy flows into the system,” or “process
X caused the kinetic energy of the system to increase.”
In both these cases, metaphorically, energy is being spo-
ken of as matter and the system as a container of energy.
(This is suggested particularly by the use of the preposi-
tions “into” and “of.”) It is no coincidence that these two
locutions are identical to examples given by Lakoff and
Johnson [25]. The authors describe similar metaphori-
cal patterns in how humans (in English at least) encode
physical processes and events as movement of substances
into and out of containers.
Physicists are aware of the limitations of their lan-
guage. When asked about what is oscillating in a quan-
tum mechanical wave, one professor responded:
Prof B: “The problem is you’re trying to shoe-
horn a phenomenon into ordinary everyday
English language, and I think the problem is
with the language, not with the phenomenon.
So, if you ask me to explain it in English, I
think English has limitations which make it
impossible to give a satisfactory explanation
in English. But, I don’t have to understand
it in English. I mean, I think I sort of know
what’s going on. At least I have realized the
limitations in English and, it doesn’t bother
C. Ontological Underpinnings
1. Introduction
It has been suggested that humans divide the world
into ontological categories of matter, processes and men-
tal states [41]. In this section we will show that this
idea can be applied to models in physics. The ele-
ments of a physical model: the objects or systems of
objects, interaction laws, force laws, state laws etc.,
may be mapped to the ontological categories of matter,
processes, and physical states. In cognitive linguistics,
Lakoff and Johnson [25] have shown that systems of con-
ceptual metaphors are based on ontological metaphors.
These ontological metaphors often give abstract concepts
an existence as concrete objects or things. To unite these
two views and systematize our linguistic analysis, we hy-
pothesize that ontological metaphors in physics language
are realized as grammatical metaphors. Functional gram-
marians have suggested [42] that the elements of a sen-
tence can be divided into participants (nouns or noun
groups), processes (verbs or verb groups), and circum-
stances (generally adverbial or prepositional phrases). In
order to unify the metaphorical and grammatical views,
we have suggested [26, 43] that grammatical partici-
pants should be mapped to the ontological category of
matter, and grammatical processes represent ontological
processes. Ontological physical states also have unique
grammatical representations, through the use of gram-
matical location.
2. A Lexical Ontology
We hypothesize that the concepts in a physical or ana-
logical model can be arranged into an ontological tree
similar to the one proposed by Chi et al. [41]. It is nec-
essary to modify Chi et al.’s ontology tree to accommo-
date one missing category: namely physical states. (See
Fig. 1.)
Etkina et al. have suggested that physical models can
be broken up into a taxonomy of (1) models of objects,
(2) models of interactions between objects, (3) models
of systems of objects, and (4) models of processes that
the objects/system undergoes [44]. In addition to their
taxonomy, we are going to suggest that there are two
classes of physical variables that describe a system or
the objects in it. These are (5) physical properties of
FIG. 1: A revised ontology tree based on [41]
objects (such as mass and charge), and (6) state vari-
ables that describe a configuration of the system (e.g.,
position, momentum) or state functions defined over a
system configuration (e.g., energy, entropy).
We will now show how Etkina et al.’s model taxonomy
can be mapped into the ontology tree shown in Fig. 1.
This mapping is shown in Table I.
Physical properties such as mass and charge should be
considered properties of objects classified in the matter
category.
The categorization of concepts in physics into an on-
tology tree (as shown in Table I), will be termed a lexical
ontology. For example, physicists generally agree that
energy is a state function, while heat and work are pro-
cesses by which energy is transfered into or out of a sys-
tem. Thus a lexical ontology refers definitions of physics
concepts into matter, processes, and states that physicists
would agree with as a community.
3. Grammar and Ontology
Although physicists can agree on the meaning of terms,
how do they represent the ontology of physics concepts
with language?
We suggest that every physical model described in lan-
guage has an ontology and that this ontology is encoded
in the grammar of the sentence. This grammatical on-
tology can be either literal or figurative (metaphorical).
If the lexical ontology matches the grammatical ontology
then the sentence is literal. If the lexical ontology does
not match the grammatical ontology of the same term in a
given sentence, then a grammatical metaphor is present.
We suggest that these metaphors may be consistently
identified by using the grammatical/ontological analysis
elaborated below. For an introduction to the methods of
functional grammar, we refer the reader to [42].
Consider for example, “John [agent] kicked [process]
the ball [medium].” Here “John” and “the ball” are gram-
matical participants, functioning grammatically as ob-
jects or matter. We also recognize that “John” and “the
ball” are naturally defined as matter in some sense. Thus
the grammatical ontology and lexical ontology match.
There is nothing metaphorical in this sentence. Consider
now for example, “heat [medium] flows [process] from the
environment to the gas.” In this sentence a physicist
would recognize heat to define a process of movement of
energy into the system (lexical ontology). But grammat-
ically “heat” is functioning as a participant, namely heat
is the matter that is flowing. In this case the grammati-
cal function of the term “heat” and the lexical ontology of
“heat” contradict each other. The sentence is therefore
metaphorical.
We are going to propose the following mapping from
grammar to the ontology tree shown in Fig. 1: Gram-
matical participants should be mapped into the ontolog-
ical category of matter. Participants can immediately be
separated into living and non-living ontological subcate-
gories: Beneficiary, agent, and medium (as it participates
in an action process, such as “a force [medium] acts [pro-
cess]”), can all be thought of as living entities. Range and
medium (as it participates passively in an event process
such as “heat [medium] flows [process]”) can be thought
of as non-living entities.
Certain parts of circumstantial elements can also be
mapped to the matter category. In the example, “. . . the
incident particles [medium] will be. . . partially transmit-
ted [process] through the potential-well region [location].”
“the potential-well region” could be classified as non-
living matter. However location also functions grammati-
cally to make ontological physical states as in “A particle
[medium] is [relational process] at coordinates (1,1,1) [lo-
cation].” This will be discussed further below.
An important type of grammatical process in the dis-
course of physics is the relational process. Relational
processes are processes of being in that they almost al-
ways include some form of the verb “to be.” Relational
processes have two modes: identifying and attributional.
The identifying mode is a reflexive relationship. For ex-
ample, “the neutrino is the lightest known particle.” It
makes sense to say “the lightest known particle is the neu-
trino.” The attributional mode denotes category mem-
bership and is not reflexive. For example: “An electron
is a lepton.”
We hypothesize that physical states (as expressed in
physicists’ language) are commonly comprised of identi-
fying relational processes where the second identifier is
missing and replaced by a grammatical circumstance of
location. Typical examples are: “The electron is in the
ground state,” “the particle is at such and such coordi-
nates.” Such sentences very often involve a grammatical
TABLE I: Table illustrating how Etkina’s model taxonomy successfully maps into Chi’s (modified) ontology
Ontological
category
Matter Process State
Ontological sub-
category
Non-living Event Procedure and Constraint-
based Interaction
Physical State
Taxonomy
element
objects system interaction
causal laws, state laws state variables, state
functions
metaphor. Ontologically location is mapped to some sort
of physical object or matter, this often conflicts with the
lexical ontology. These grammatical metaphors corre-
spond directly to the ontological metaphors of Lakoff and
Johnson in choice of preposition: “in” implies container,
“at” implies point location in either time or space, “on”
implies surface. We believe that it is also no coincidence
that these statements have a grammatical structure iden-
tical to those of mental states. For example, in English
we say, “I am in love,” or “I am in trouble,” or “I am in
a state of confusion” etc. It seems to us that physicists
have borrowed this metaphor wholesale and blended it
with the notion of a physical state, to create a way of
speaking about physical states.
Ontological processes that describe the behavior of a
physical system, are realized in speech by grammatical
material processes. Relational processes realize either
physical states as shown above, or denote some compo-
nent of the model in the sense of category membership.
In grammar there are two types of material process: ac-
tion and event. These two types of process can be used
to distinguish between living and non-living matter.
The entire mapping from grammar to ontological cat-
egory is summarized in Table II below.
TABLE II: Summary of the mapping between grammar and
ontological category
Grammatical
function
Ontological category
(If X functions grammat-
ically as. . . )
−→ (. . . classify X ontologi-
cally as. . . )
Agent −→ Matter:living
Beneficiary −→ Matter:living
Medium (action process) −→ Matter:living
Medium (event process) −→ Matter:non-living
Role −→ Matter:non-living
Objects in Location −→ Matter:non-living
Process −→ Process
Manner −→ Process
D. Summary
The theoretical framework is summarized in Fig. 2 be-
Consider, for example, the caloric theory of heat. This
theory of thermodynamics began as an analogy to a
FIG. 2: Summary of the role of analogy, metaphor, ontology
and grammar
weightless fluid in the late eighteenth century. Over time
the elements of this theory became encoded in the lan-
guage of physics as a conceptual metaphor. For example,
physicists today still say “heat flows from object A to ob-
ject B,” and talk about the “heat capacity” of an object.
Phrases and sentences such as these are evidence of the
conceptual metaphor heat is a fluid in physicists’ lan-
guage. For physicists, speaking about heat is a fluid is a
productive mode of reasoning as long as there is no work
being done on or by the thermodynamic system. The ap-
plicability and limitations of talking about heat as a fluid
are communally well understood. The analogy between
heat and a fluid has an underlying ontology of matter
(the heat fluid), processes (the movement of heat from
one object to another), and states (the amount of heat
in an object — indicated by the object’s temperature).
This ontology is encoded in the grammar of each sen-
tence used to speak or write about the thermodynamic
system. In the example, “heat flows from object A to
object B,” heat is a grammatical participant, while the
grammatical process is “flows”. Object A and B are parts
of grammatical location. Implicitly, the amount of heat
in object A or object B indicates the current state of the
system. In the modern thermodynamic model, the onto-
logical matter is the atoms or molecules in the system,
the processes that the system undergoes are heating and
work (processes of energy transfer), and the state of the
system is represented by the energy or entropy of the sys-
tem for a given configuration of the molecules. Note how
the modern ontology is in direct conflict with the caloric
ontology of thermodynamics. Speaking about heat as
matter is therefore a grammatical metaphor. It is gram-
matical metaphors like this that underpin the conceptual
metaphor heat is a fluid.
E. Student Difficulties, Student Learning
The central question of our paper is: What is the in-
terplay between the linguistic representations that physi-
cists use and students’ learning and students’ difficulties?
We will narrow this down to two hypotheses regarding
the role of language and learning in physics. These are
elaborated in Sections II E 1 and II E 2 below.
1. Student Difficulties Interpreting Metaphors
Students struggle to see the applicability and limita-
tions of analogies that they encounter. We suggest the
same applies to metaphorical language that they hear
and read. To comprehend a metaphor people construct
an ad hoc category [32, 33]. This means that a state-
ment of the form “X is Y” has to be interpreted through
the formation of a new shared category (an ad hoc cate-
gory) of which Y is a prototypical member. For example,
to comprehend a metaphor such as the electron is
a smeared paste, the reader has to come up with an
ad hoc category shared by both entities. A physicist who
understands the quantum mechanical behavior of an elec-
tron, might suggest an ad hoc category of “things that
don’t have a well-defined location.” There is no guaran-
tee that a student will come up with the same classifica-
tion. We hypothesize that students often come up with
an ad hoc category that is inappropriate to a given situa-
tion. This inappropriate categorization is at the heart of
their difficulties. These difficulties may manifest them-
selves as “misconceptions” or student difficulties. We
predict that students will overextend and misapply key
aspects of metaphorical systems in physics. Instances
where metaphors are overextended or taken too literally
will be connected with their faulty reasoning.
In order to test these ideas it is first necessary to iden-
tify if there are really coherent systems of conceptual
metaphors in the way physicists speak and write. In Sec-
tion III we will show some of the interview data with
physics professors that lead us to see that this view of lan-
guage was really applicable to the discourse of physics. In
Section IV we will consider examples of student difficul-
ties in QM that we can explain as examples of metaphor-
ical overextension.
2. Students’ Ontological Confusion
Previously, Chi et al. have shown that many student
“misconceptions” are based on students’ incorrect onto-
logical classification of physics concepts [41, 46, 47]. For
example, physicists classify heat is a process, but students
reason with it as if it were matter.
We want to propose an extension of this idea. From our
data it appears that physicists reason about physics by
co-ordinating multiple descriptions of a particular phe-
nomenon. These descriptions may possess different or
conflicting ontological properties. For example, there
are times when physicists talk about QM phenomena
in terms of waves (a process description) whereas there
are other times when physicists prefer to talk about a
QM phenomenon in terms of particles (a matter descrip-
tion). Physicists are good at co-ordinating these different
and sometimes conflicting descriptions. Physicists under-
stand when a wave or a particle description work and how
use them appropriately in their reasoning.
Students learn these descriptions by listening to and
reading what their teachers say and write. Our sec-
ond hypothesis is that students are failing to co-ordinate
appropriately the many different descriptions that they
learn from physicists’ language. For example, physicists
often describe the “potential energy graph” in QM in
terms of physical objects (well, barrier, etc.), endowing
the graph with the properties of a physical object. Stu-
dents, hearing this language, also learn to think of the
graph as a physical object. However, students are often
unaware of when this picture is appropriate or inappro-
priate. Thus students may attach inappropriate ontolog-
ical properties to the idea of the graph as a physical ob-
ject. We hypothesize that this process leads to patterns
of student reasoning that researchers sometimes interpret
as “misconceptions”.
In Section IV we will test this hypothesis by analyz-
ing an example of students solving a QM problem and
consider several studies from the PER literature.
III. METAPHORS IN QUANTUM MECHANICS
We will trace two metaphorical systems in QM from
their origins as analogies through to modern language
that physicists use to speak and write about their
ideas. These two systems are (1) the potential well
metaphor, and (2) the Bohmian metaphor.
Both grammatical and metaphorical analyses will serve
together to illustrate a number of claims made in Sec-
tion II. The claims are: (1) Coherent systems of
metaphors exist in physicists’ language. (2) Systems of
metaphors encode historical analogies. (3) The language
encodes a representation or representations of a phys-
ical model that has an underlying ontology of matter,
processes and states. (4) Physicists use these linguistic
representations to reason productively about certain phe-
nomena.
The data for the linguistic analysis will come from two
sources. The first is the interview study with physics
professors referred to in Section II. These professors were
all native English speakers We asked them to describe
and explain various ideas in QM such as the Heisenberg
uncertainty principle, or how they would respond to a
student who asked, “what is oscillating in a QM wave?”
The interview study consisted of five subjects. The full
set of interview questions may be obtained by request
from the authors. The second source of data is a selection
of QM textbooks [19, 20, 21, 22, 23, 24].
A. The Potential Well Metaphor
1. Original Descriptive Analogy
“Because of the Pauli exclusion principle, the
electrons must be spread over the available
states; but they settle down to the states of
lowest energy, so that as more electrons are
added, the energy levels in the band fill up
like a bucket fills with water.” [48]
In this example Peierls makes the analogy explicit. The
way he uses it shows that this analogy has a descriptive
2. Analysis of Modern Language
Grammatical and ontological analysis: When
physicists speak of “potential well” and “energy level,”
they give energy an existence as water. When physicists
speak about quantum particles “leaking through a
barrier,” they give the quantum particles an existence
as water. When physicists speak of a “potential well,”
“potential step,” “potential barrier,” “confinement,”
“trap” etc. . . they give the potential energy graph an
existence as a physical object. This ontology is encoded
in the grammar. Samples of textbook writing and
physicists’ talk from interview data and accompanying
grammatical analysis are provided in Table III.
From the data we have studied, this selection of talk
and writing of physicists (Table III) is representative
of the type of language associated with the potential
well metaphor. One can see a clear pattern of gram-
mar that can be mapped to the ontological categories of
matter and processes. This is shown in Table IV below.
The grammatical analysis can tell us more than what
the objects and processes are in the metaphorical model.
It shows us that the potential well metaphor consists of
two objects, the particle/wave function, and the poten-
tial energy graph, that function as separate grammatical
participants. They interact with each other via a number
of possible processes such as “tunnel through,” “reflects,”
and so on. Thus the common grammatical structure of
the potential well metaphor contradicts the conven-
tional view of the potential energy as a property of the
particle or the system.
Metaphorical analysis The ontology encoded in the
grammar describes the basic objects and processes of the
physical model. To understand more subtle properties of
these objects, and their interactions, we need to apply a
metaphorical analysis. For this, we need to identify the
base of domains of various analogs that go into making
up the potential well metaphor.
In this section we will analyze an additional sample of
clauses and sentences from a selection of popular intro-
ductory quantum mechanics textbooks [19, 20, 21, 22,
23]. We will identify each metaphor that makes up the
metaphorical system, and present sample examples of its
use by physicists. Additional examples may be found
in [26].) Where necessary, we will identify the analogi-
cal base from where the words have been “borrowed” to
create the metaphor.
• Metaphor: The potential energy graph is a
physical object or physical/geographical
feature.
Examples:“The perfectly rigid box, represented
by a rectangular potential well with infinitely high
walls, is an ideally simple vehicle for introducing
the mathematics of quantum systems.” [21]
“Scattering from a ‘cliff’.” [23]
“. . . even for a total energy of the particle less than
the maximum height of the potential hill. . . ” [19]
“What are the classical wave analogs for particle
reflection at a potential down-step and a potential
up-step?” [21]
Base: The words “box,” “well,” “hole,” “cliff,” and
“hill” are borrowed from the category of physical
objects or physical/geographical features.
• Metaphor: The previous metaphor the poten-
tial energy graph is a physical object or
physical/geographical feature entails an-
other metaphor: The “walls” of the well or bar-
rier correspond to a physical height above the
ground. In other words, energy is a vertical
spatial dimension in the Earth’s gravitational
field. The potential energy graph is a phys-
ical object or physical/geographical fea-
ture metaphor builds on this spatial metaphor.
Examples:“It is instructive to consider the effect
on the eigenfunctions of letting the walls of the
square well become very high. . . ” [20].
Prof A: “. . . your zero point energy is going to go
up and up.”
“. . .ψ1, which carries the lowest energy, is called
the ground state. . . ” [23]
Base: The words “high,” “up,” “lowest,” and
“ground” all suggest an analogy between the verti-
cal axis of the potential energy graph and a vertical
spatial dimension on the Earth’s surface.
• Metaphor: The potential energy graph is
a container. The potential energy graph “con-
tains” or “traps” either the wave function, the par-
ticle or the energy of the particle.
TABLE III: Samples of physicists’ speech and writing for grammatical analysis.
Sample of physicist’s speech or writing Simplified exerpt with analysis
“In both cases, a classical particle of total energy E. . .moves
back and forth between the boundaries.” [21]
a classical particle [medium] moves back and forth [pro-
cess:event] between the boundaries [circumstance:location].
“. . . when you have a confined system, . . . [the width of the
box is] going to set the scale for what . . . the magnitude of
the energy is, so as you confine it [the particle] more and
more, your zero point energy is going to go up and up.” -
Prof. A, interview study
. . . your zero point energy [medium] is going to go up and up
[process:event].
“. . . it has been seen that potential barriers can reflect parti-
cles that have sufficient energy to ensure transmission clas-
sically.” [19]
. . . potential barriers [agent] can reflect [process:event] parti-
cles [medium]. . .
“This [wave] packet would move classically, being reflected
at the wall. . . ” [22]
This wave packet [medium] is reflected [process:event] at the
wall [circumstance:location].
“The α-particle then ‘tunnels through’ the barrier. . . ” [19] The α-particle [medium] then ‘tunnels through’ [process] the
barrier. . . [range]
“. . . they [α particles] start out with the energy E inside the
nucleus and ‘leak’ through the potential barrier.”[24]
α particles [medium] leak through [process:event] the poten-
tial barrier [range].
“. . . the phenomenon of tunneling. . . allows the particle to
‘leak’ through any finite potential barrier. . . ” [23]
The particle [medium] leaks through [process] any finite po-
tential barrier [range].
TABLE IV: Ontology of the potential well metaphor
Matter Process
QM/classical particles, wave packet,
energy, energy walls, energy barrier,
potential barrier, barrier
moves, reflect(s),
tunnels through,
leaks through
Examples: “The exponential decrease of the wave
function outside the square well for the second
energy state is less rapid than is the correspond-
ing decrease for the lowest energy state as indi-
cated. . . ” [21]
“Inside the well where V (x) = −V0. . . ” [23]
“. . .bound [energy] states in the. . . well” [21]
Base: Words such as “well,” “confined,” “bottle,”
and “bound” all suggest an analogy to some sort
of container. Physicists also make a distinction
between “in/inside” and “outside” the well: Such
adverbial phrases also indicate the presence of the
container metaphor.
Various other elaborated metaphors are built on this ba-
sic set. Examples of their usage may be found in [26].
• The potential energy graph is a barrier.
• The potential energy graph is a hard
container/barrier or the potential energy
graph is a semi-hard container/barrier
• The particle, the wave packet, and the energy are
all given an ontological status of matter. More
specifically:
– QM particles are hard objects.
– The wave packet is a soft or breakable
object
– QM particles are a fluid
– The energy is a fluid
See Table V for a summary of the metaphorical map-
ping from the domain of physical/geographical features
to the domain of quantum systems that involve an inter-
action between two or more objects.
3. Productive Modes
How do physicists piece together the gram-
mar/ontology of the potential well metaphor?
How do they use the associated imagery to reason
productively about quantum systems? From the dis-
course of professors and textbooks we have identified the
presence of productive modes for the potential well
metaphor. We present five examples below:
Squeezing: Squeezing the walls of the well forces the
water upwards, thereby raising and spacing out the “en-
ergy levels.”
Example: Prof. A: “. . . when you have a confined sys-
tem, . . . [the width of the well is] going to set the scale for
TABLE V: Summary of the metaphorical mapping between
the base domain of physical/geographical features and the tar-
get domain of interacting QM systems
Base domain Target domain
Physical/geographical
features
Interacting QM systems
Physical or geographical
features
→ Potential energy graph
Vertical height of physi-
cal/geographical feature
→ Magnitude of energy at a
point/region on the poten-
tial energy graph.
Hardness or softness of a
→ “Height” of the potential en-
ergy graph
Container with top face
→ “Trapping” of QM particles,
“bound” states
Billiard ball → QM particle in some
circumstances
Soft or breakable objects → QM wave function or wave
packet
Fluid → QM particle in some circum-
stances, or the energy of the
particle/system
Ball bounding off a wall → Reflection of QM particle
Tunneling/penetration → Process by which a QM
particle “passes through” a
seemingly solid “barrier”
Leaking → Process by which a QM par-
ticle “escapes” from a QM
“container”
what . . . the magnitude of the energy is, so as you confine
it more and more, your zero point energy is going to go
up and up.”
Stacking: Matter takes up space. Filling up the
well/bucket can be used to understand the behavior of
fermions.
Example: We already observed Peierls make this anal-
ogy explicit[48]. The following is an example of Prof. A.
using it: “if you have fermions then. . . you have to keep
stacking the fermions into levels which get more and more
elevated in energy. . . ”
Tunneling/leaking: The potential energy graph be-
haves as a physical container or barrier, preventing the
escape of the particle. This leads to the ideas of “tunnel-
ing” or “leaking.” When reasoning productively, physi-
cists recognize that a higher or wider barrier means less
probability of tunneling.
Examples: Feynman et al. write: “. . . they [α-particles]
start out with the energy E inside the nucleus and ‘leak’
through the potential barrier.”[24]
Griffiths writes: “If the barrier is very high and/or
very wide (which is to say, if the probability of tunneling
is very small), then the coefficient of the exponentially
increasing term (C) must be small. . . ” [23].
Reflecting/scattering: The wave function or particle
is reflected by or scatters off a hard barrier.
Example: “For this reason, the rectangular potential
barrier simulates, albeit schematically, the scattering of
a free particle from any potential.” [22]
A way of speaking: It is difficult to come up with
realistic physical systems of quantum mechanical parti-
cles without resorting to lengthly descriptions. (See [21]
for examples.) By separating the QM particle from its
potential energy graph, physicists are able to talk easily
about the particle interacting with an external object (its
own potential energy graph).
Example: During one of the interviews we asked a pro-
fessor to describe the process of trapping and cooling
atoms to absolute zero.
DTB: “Are the atoms going to jump out, are
you not going to be able to trap them?”
Prof. E: “No, of course not, you’d just go
down to the lowest eigenstate. I mean, I don’t
know how they were trapped in the first place,
but suppose you had them in a square well for
example.”
4. Summary
We have tried to illustrate how grammar and metaphor
work together to encode the features of a particular de-
scriptive model. Each aspect is necessary and the gram-
matical and metaphorical analysis together serve to illu-
minate features that each individual analysis cannot do
on its own.
Fig. 3 presents a visual summary of how the po-
tential well metaphor is structured. The poten-
tial well metaphor is an example of a metaphor-
ical system made up of three ontological metaphors,
two of which are encoded in the grammar (the po-
tential energy graph is a physical object or
physical/geographical feature, and the parti-
cle/wave function/energy is a physical object
or matter), and one which can only be identified by
looking at the imagery (energy is a vertical spatial
dimension). Other metaphors such as the potential
energy graph is a container or the potential en-
ergy graph is a hard barrier build on and elab-
orate this ontology. At the sentence level, we can see
how productive modes of reasoning are formed by intro-
ducing grammatical processes through which the particle
or wave function interacts with its own potential energy
graph. These productive modes are squeezing, stacking
tunneling/leaking, and reflecting/scattering. Physicists
also use the potential energy graph is a physical
object metaphor as a substitute term (or metonym) for
the actual physical QM system.
FIG. 3: Summary of the metaphorical system and its usage by physicists.
B. The Bohmian Metaphor
1. Introduction
The potential well metaphor, as a linguistic repre-
sentation, has many of the characteristics of a physical
model as described by Etkina et al. [44]. The language
describes objects with properties, and processes by which
those objects interact with each other. In contrast, the
Bohmian metaphor has almost none of those character-
istics. It seems to exist in the language of physics solely
as a way of speaking. It is an interesting case because
it is easy to identify the metaphor, but not the original
analogy. Therefore, in this section, we will present the
linguistic analysis first and the study of the analogy on
which it is based, second. Although we have called it
the “Bohmian” metaphor in honor of David Bohm, who
advocated the Bohmian interpretation of QM, the entry
into the language of physics can be traced back much
earlier than this.
2. Modern Language
The Bohmian metaphor is identified in language by
words and phrases that suggest that the wave function or
quantum state is a container that contains the quantum
mechanical particle. There are only two metaphors that
make up the Bohmian metaphor:
• Metaphor: The wave function/quantum
state is a container.
Examples: Noun groups such as “wave packet”
or “envelope function” indicate an analogy to a
container.
• Metaphor: The QM particle is a physi-
cal object contained inside the wave func-
tion/quantum state.
Examples: This is suggested by prepositional
phrases such as “in the ground state” in sentences
such as “The electron is in the ground state.”
Connection to grammar: In the Bohmian
metaphor the wave function or quantum state is
conceived of as a container that has a particle as a
separate entity inside it. The language is based on two
sources. The first source is an analogy to Einstein’s
ghost field idea (see Section III B 3 below) but the second
source is language itself. Cognitive linguists hypothe-
size that mental states are spoken about in language
metaphorically as containers [25]. For example, if one
is depressed one can say, “I am [relational process] in
a state of depression [location].” Such statements seem
to all have the same grammatical structure, namely a
relational process followed by circumstance of location. It
seems as if ontological physical states are expressed by an
identical grammatical structure: such as, “the electron
is [relational process] in the ground state [location].”
It seems as if physicists have unconsciously borrowed
this grammar that expresses mental states in every day
experience and used it to express physical states in
physics. As mentioned in Section II, the metalingual
apparatus that we have for realizing physical states in
language appears to be extremely limited. The states
are locations metaphor, supported by this unique
grammatical structure, is one of these limited means
of expression. A statement about the physical location
of an object within another object would be classified
in the ontological category of matter if taken literally.
Metaphorically a statement such as “the electron is in
the ground state,” is a statement about the energy of
a quantum system, and physicists recognize energy as
a state function. There is a clear ontological conflict
between the literal interpretation of the statement
and the meaning that is intended. This leads us to
hypothesize that such statements will cause students
confusion and may lead to difficulties.
3. The Original Analogy
In the case of the potential well metaphor, the
analogy on which it is based, is relatively well under-
stood. The Bohmian metaphor is easy to identify, but
the original analogy is not well known. If our frame-
work is correct and language is built on analogy then an
original analogy should exist in the mainstream QM lit-
erature. We started searching the original QM papers in
the hope that we would find some explicit reference to
the idea that the wave function could contain the parti-
cle inside it. Remarkably, we found such a reference in a
paper by Max Born, published in 1926 [16].
“Neither of these two views seem satisfactory
to me. [Heisenberg’s interpretation of the
wave function and the Schrödinger/deBroglie
interpretation of the wave function] I would
like to attempt here a third interpretation and
test its applicability to collision processes. I
thereby pin my hopes on a comment of Ein-
stein’s regarding the relationship between the
wave field and light quanta. He says roughly
that the waves may only be seen as guid-
ing [showing] the way for corpuscular light
quanta, and he spoke in the same sense of a
“ghost field.” This determines the probability
that one light quantum, which is the carrier of
energy and momentum, chooses a particular
[definite] path. The field itself, however, does
not have energy or momentum.” [16] [Trans-
lation by D.T.B.]
There are several remarkable features about this pas-
sage from Born:
• Firstly, it lays out the Bohmian interpretation of
quantum mechanics twenty-five years or more be-
fore Bohm proposed the same idea, and one year
before deBroglie’s attempt at a “pilot wave” the-
• Secondly, when Born says “Neither of these two
views seem satisfactory to me,” he is referring
to (1) the Heisenberg interpretation of QM which
Born describes as “an exact description of the pro-
cesses in space and time are principally impossi-
ble,” and (2) the Schrödinger/deBroglie interpreta-
tion which Born summarizes: “He tries to construct
wave groups which have relatively small dimensions
in all directions and should, as it seems, directly
represent moving corpuscles.” Born is cautioning
against overly literal interpretations of (1) an anal-
ogy to a classical particle (Heisenberg’s approach),
or (2) an analogy to a physical wave (Schrödinger’s
approach). Born suggests that both views lead to
untenable positions in the physical interpretation of
QM and introduces a third model which is essen-
tially a hybrid of the wave and particle analogies.
Born makes, an analogy to Einstein’s interpreta-
tion of light waves and light quanta and applies it
to particles with non-zero mass.
Born’s mode of reasoning appears to be metaphor-
ical as well as analogical. He makes an analogy
to Einstein’s view of the electromagnetic field as a
ghost field, but he does not suggest that the wave
function is “like a guiding field.” Rather, he ex-
presses Einstein’s idea directly as “. . . the waves
may only be seen as guiding the way for corpus-
cular light quanta. . . ” [our emphasis]. For Born to
interpret the wave function as a probability distri-
bution, he felt it necessary to blend together a wave
picture and a particle picture with real particles
who have definite trajectories determined proba-
bilistically by the wave function. Lakoff and Núñez
refer to such a mental construct as a metaphorical
blend [28] after the conceptual blend of Fauconnier
and Turner [31].
• Thirdly, Born is aware of the limitations of the
metaphorical picture he has introduced. In blend-
ing a wave and particle picture into a model that
looks and feels like a statistical ensemble, Born cau-
tions about taking this “Bohmian” picture too lit-
erally when he writes: “However, the proposed the-
ory is not in accordance with the consequences of
the causal determinism of single events.” [16]
4. Productive Modes
One of the difficulties with QM is the question of how
to speak about quantum processes meaningfully. We sug-
gest that the Bohmian metaphor permits a partial so-
lution to this problem. Although Born’s suggestion (in-
tepreting the wave function as a pilot wave) never made
it to the mainstream of physics, the associated language
is now ubiquitous and used productively by physicists as
we will show in the following example:
D.T.B.: “. . . if you wanted to think about how
an electron propagates. . . It wouldn’t be sen-
sible to talk about it as a wave, you would
think more as a particle?”
Prof C: “. . . you can think of it as a plane
wave. Yeah, . . . in an envelope function
which makes it into a wave packet.”
More examples may be found in [26].
IV. STUDENT DIFFICULTIES
A. The Potential Well Metaphor
A group of four junior students in their first QM course,
were video taped while working on their QM homework
problems. All students were native English speakers.
The discussion we present is centered around a problem
from French and Taylor[21]. The question was: “What
are the classical wave analogs for particle reflection at
a potential down-step and a potential up-step?” Notice
here the potential well metaphorical system serving
a specific function: namely, it describes the shape of the
potential energy graph (“potential down-step”).
S1: Well, there wouldn’t be reflection in par-
ticle physics on a down-step right? Or even,
I don’t think even on an up-step. . .
S3: No, there’s reflection on an up-step, total
reflection.
S1: Not classical though, right?
S2: Not if its less than the energy though.
S1: It just slows it down.
In this opening exchange we can observe S1 talking at
cross purposes with S2 and S3. S2 and S3 seem to be
imagining a classical particle approaching the step and
bouncing back (later dialogue show that they do not re-
ally shift from this literal view of the situation), while S1
seems to be thinking of a wave approaching with energy
greater than the energy of the step. As we see later, S1
is reasoning from picture of a surface water wave passing
over a step in a river or sea bed.
S1: Not quite sure what the wave analogs
would be. If I had to guess I’d say it would
be like sound, like those things that male
cheerleaders have, like big cones.
S4: Megaphones?
S1: Yeah. ’Cause I think, you
know,. . . basically a step up or step down in
resistance. But I am not quite sure what we
are supposed to say about that.
This is the first example of an analog from S1. It is
interesting that S1 sees the key as a change in resistance
(at the end of the first exchange S1 says “It just slows
it down”), yet he still is the one who proposes a phys-
ical form (consistent with the ontology of the graph as
a physical object) surrounding the medium rather than
a change in the medium itself (which would represent a
more obvious change in resistance for the wave).
S2: So they’re saying that there would be re-
flection on a potential up-step like a. . .
S1: Yeah, just like a sound, or a water wave
or something.
S1: Um, well ’cause I know on a potential up-
step,. . . like if you just had. . . water and you
had, you know, deeper part and a shallower
part, and you had a wave, some of it would
reflect back.
Here S1 applied the metaphor of a physical object
again, and proposes a second analog based on the physi-
cal form of the graph rather than a change in “density”
or “tension” of the medium. Actually, a physical step on
a river bed could be a valid example if S1 connected it to
a model of how the resistance experienced by a surface
wave attenuates with the depth of the water. He does
not, and this explains his uncertainty below.
S1: So that’s not too hard to see. But like, I
would guess that the same thing would hap-
pen if you had a down-step, but that’s not
something like I really, I could vouch for. Like
I think they’re looking for stuff that like most
people know.
S2: Is that what its saying? Its coming at
it with every energy, like continuous energies,
like around the step?
S2’s statement is interesting. The use of “at” and
“around” are examples of grammatical location and sug-
gest the metaphor: the step is a physical object.
S1 shows he is still on the right track when he says:
S1: I think they’re just asking for like,
examples from. . . in real life from when a
wave. . . goes into a space of less resistance and
has reflection back.
S4: So in classical what would happen at a
potential down-step?
S1: A potential down-step?
S2: It would just keep going. . .
S1: . . . It would just speed up. At a potential
up-step it would just slow down.
1. Discussion
One alternative hypothesis to explain the difficulties
presented above could be that the students are unable
to interpret the physical meaning of the potential en-
ergy graph or are simply not understanding the situation.
However, S1’s ability to interpret potential energy graphs
correctly and articulate the key to the analogy discounts
this hypothesis. The data show that his inability to come
up with a productive analog must be based on other fac-
tors. Our framework explains how S1 is distracted by ap-
plying an overly literal interpretation of the potential
well metaphor in an inappropriate situation. Possibly,
a way of talking (i.e., describing the potential graph as
a “step”) is affecting students’ reasoning. Our analysis
(Section IV above) shows that the students in this group
are searching in the category of “physical objects” for an
analogy, in accordance with the underlying ontological
metaphor the potential energy graph is a phys-
ical object rather than searching in a more produc-
tive category. Other researchers have also noticed that
QM students tend to pick 2-d gravitational analogs when
asked to come up with physical examples of 1-d potential
energy graphs [49, 50].
As a control we posed the same problem to the profes-
sors in the interview study. They all responded that an
analogy of an electron beam scattering off of a potential
down step is light traveling from a medium with greater
index of refraction to a medium with a lesser index of
refraction. When asked why changing optical media was
a good analog, most were unable to explain, but contin-
ued to elaborate their answer. Only one professor was
able to explain why this was a good analogy. Prof. E:
“I know because we’ve thought about these things before
and its just been classified in that category.” This state-
ment suggests that physicists are able to automatically
search for an analog in a category of analogous processes
rather than analogous objects. It may also suggest that
physicists’ ideas have become so tightly bound into larger
conceptual units that professors are unable to break down
their reasoning into smaller parts again.
We have shown how physics professors can use
metaphorical systems to reason productively in certain
situations while students take the same representation
and apply it too literally and inappropriately in other
situations. Strange ideas like the megaphone make sense
if we understand the underlying ontology of the graph,
spoken of as a physical object. We think that the ex-
ample of student discourse presented above is a typical
example of students’ difficulties arising from linguistic
representations.
2. “Robust Misconceptions” Related to the Potential
Well Metaphor
Are there “robust misconceptions” in QM? The char-
acteristics of a robust misconception are that it must be
(a) present before instruction, (b) common to a signifi-
cant percentage of students in a particular class, and re-
producible in form and structure across different classes
at different institutions in different contexts, and (c) re-
sistant to instruction. (See [51] for example.)
Although research on students’ understanding of QM
is in its infancy, it appears that students do have spe-
cific difficulties that have the characteristics of a robust
misconception. One emerging example is presented in
Table VI. It has been observed that students think that
a QM particle loses energy when it tunnels through a bar-
rier. McKagan et al., who studied this example, freely
use the word “misconception” in their paper [52].
A consistent pattern of reasoning is presented in Ta-
ble VI. This pattern contains the following two elements:
(1) It takes energy for a particle to tunnel through a
barrier. (2) Making the barrier wider or higher means
that the particle loses more energy/expends more effort
when tunneling through it. Morgan et al. speculate that
the difficulty may come from either (a) intuitive classical
ideas about a particle passing though a barrier, or (b)
physicists tend to draw the potential energy graph and
the wave function superimposed. Thus a decaying wave-
function amplitude may be confused with a decrease in
energy.
McKagan et al., however, noticed something interest-
ing in their study. In interviews, they discovered that
students do not see the potential energy graph as repre-
senting the potential energy of the particle in question.
They see it rather as some external object with which
the particle interacts. The authors describe an example
from their interviews:
“When pressed, he said that the ‘bump’ was
‘the external energy that the electron inter-
acts with’ and insisted that it was not the po-
tential energy of the electron itself, in spite of
the fact that it was explicitly labeled as such
in the previous question.”
The authors speculate that statements like “a particle in
a potential” may be the cause of this problem.
Our analysis supports this idea and provides an expla-
nation for the underlying causes of this student difficulty.
The problem is much more widespread than just phrases
like “a particle in a potential.” As we pointed out in
Section III A 2, many statements that fall under the cate-
gory of the potential well metaphor, tend to separate
the particle or wave function from its potential energy
graph in the grammar of the sentence. Most often the
particle/wave function functions grammatically as the
medium while the potential energy “barrier” functions
as either the range, or circumstance of location. The two
grammatical participants then interact with each other
by a grammatical process such as “tunnels through” or
“is reflected.” We hypothesize that the language is the
primary source of the students’ model. Graphical repre-
sentations (such as the superposition of the energy graph
and the wave-function) and classical intuitions build on
and extend this basic model, leading to the idea that
energy is lost in the tunneling process.
TABLE VI: Selected examples of the “exhaustion” misconception: Summary from three studies.
Authors’ summary and explanation Sample student responses used to justify this
explanation.
Lei Bao [49] interviewed ten students over two semesters.
Three responded with the incorrect idea that a quantum
mechanical particle loses energy when it tunnels through a
potential barrier.
Bao observed that all three students gave similar re-
sponses. Mike: “. . . less energy so the amplitude will be
reduced,. . . Amplitude is reduced because energy is lost in
the passage [our emphasis]. . . ”
Jeffrey Morgan et al. [53] found that all six students that
they interviewed thought that the particle lost energy when
it went through a potential barrier. Two of the students had
completed a senior level QM course and four had completed
a sophomore level introductory QM course
Selena: “Uh, because it requires energy to go through this
barrier.”
Jack: “. . . when the particle of some . . . energy, encounters a
potential barrier, there is a possibility. . . that a particle will
actually just go straight on through, losing energy as it does
so, and come out on the other side. . . at a lower energy. . . ”
McKagan et al. [52] gave a conceptual test to a group of
engineering majors (N = 68) and physics majors (N = 64)
after they had completed a modern physics course. One of
the questions probed students’ understanding of tunneling
processes. On this question 24% of the engineering majors
were able to answer correctly and 38% of the physics majors
were able to answer correctly.
No interview samples were provided, but the authors sum-
marize the student responses as follows: “all students who
selected answers A, B, or E [more than 50% for both en-
gineers and physicists] argued that since energy was lost in
tunneling, making the barrier wider and/or higher would
lead to greater energy loss.”
3. Summary
The example of the potential well metaphor illus-
trates how the language used to describe certain QM sys-
tems may pose extraordinary difficulties, especially if stu-
dents are not aware of how and why metaphorical terms
are being used. The metaphorical language, grounded in
the classical world, may encourage students to associate
extra (classical) properties to the QM system as they try
to coordinate these new representations with their prior
understanding of the world. These over-extensions of the
representation seem to be the source of their difficulties.
B. The Bohmian Metaphor
As part of our study, two senior undergraduate physics
majors (in their second QM course) agreed to be video-
taped while working on their QM homework together.
Both were native English speakers. In this particular ses-
sion S1 and S2 were working on a problem worked out in
class by the lecturer that they did not understand. The
question may be expressed as follows: “Given an electron
in the ground state of an infinite square well of width L.
The walls are suddenly moved apart so that the width
of the well becomes 2L. What is the probability that the
electron is in the ground state of the new system?”
The two students working on the problem understood
the sudden approximation, they calculated the overlap
integral and got a numerical answer which was reason-
able. Then S1 stopped and pondered that his answer
made no sense. He argued that his answer should be
zero. A discussion with the observer (D.T.B.) followed.
S1: But I am still confused about what I
was. . . saying about if there is a probability
FIG. 4: Wave function of the electron in the sudden approx-
imation
that it is in the [sic] first ground state — it
seems to say that the particle can be where
it is not.
D.T.B.: Why do you say that?
S1: Because we know that the wave function
looks like this [points to a sketch similar to
Fig. 4] — Oh, so its not the probability of it
being in the ground state really. . . I think the
probability is really. . . I mean, we know that
its in this state [points to sketch similar to
Fig. 4] so it can’t be in the ground state. So
it’s zero [the probability].
The discussion circled around this theme for some
time. S1 was concerned that if the particle was “in the
ground state” of the new well, it would permit the parti-
cle to exist outside of the [-L/2,L/2] region of its initial
wave function. The wave function limits where the par-
ticle can be, but to say the electron is “in the ground
state of the new well” does not suddenly permit it to ex-
ist outside of the [-L/2,L/2] region; it is simply a state-
ment about measuring the energy of the electron. We
believe that the linguistic framework we have developed
provides both a reasonable and parsimonious explana-
tion for S1’s difficulties. The prepositional phrase,“in the
ground state,” is functioning grammatically as a location.
S1’s argument, that the probability should be zero, draws
specifically on the location metaphor. He says, “it [the
original question] seems to say that the particle can be
where it is not.” This statement suggests that he is view-
ing the question as a question about the location of the
particle. In other words, he is interpreting the phrase “in
the ground state” literally rather than figuratively.
This difficulty with the Bohmian metaphor remains
undocumented in the physics education research liter-
ature. However, a physics professor who teaches un-
dergraduate quantum mechanics, reported in a private
conversation that he observed the identical difficulty
amongst his students with the same sudden approxima-
tion problem.
V. CONCLUSION
We have shown that coherent systems of metaphors
exist in physicists’ language. We have shown that physi-
cists use these metaphorical systems in their language
to speak and reason productively about QM systems.
They are able to invoke many different metaphorical sys-
tems, sometimes with apparently conflicting ontologies,
depending on the situation they are trying to describe.
At the same time, physicists appear to understand the
applicability and limitations of their metaphorical lan-
guage in each situation. We have also shown how these
metaphorical systems can be identified with systematic
use of both grammatical and metaphorical analysis. And
we have shown how the elaborated metaphors build on
the underlying ontology encoded in the grammar.
In some cases, it seems that physicists have appro-
priated conceptual metaphors from language to express
their ideas. The example with Born and the Bohmian
metaphor shows how a new idea in physics comes out of
a blending of older ideas into a metaphorical blend. Like-
wise the final product of the language is (in this case) a
blend between an analogy to Einstein’s ghost field and
also already existing structures in language that are nor-
mally used to describe ontological mental states.
We have presented two case studies of groups of stu-
dents struggling with and being confused by overly lit-
eral interpretations of the metaphorical language they
encounter in QM. The context of QM is particularly con-
vincing because it is difficult to argue that students enter
their QM course with preconceptions or misconceptions
about QM based on personal experience. Many of the
difficulties observed, appear after instruction. It seems
more plausible to hypothesize that these difficulties are
related to the way in which physical ideas are presented
during instruction itself.
We have presented one example (the exhaustion mis-
conception) of a documented common conceptual diffi-
culty that students have with QM and how we can ac-
count for their näıve model with the linguistic framework
we have developed. Physicists understand that higher
barrier means a slower rate of tunneling or leaking. In
contrast, students think that the particles get tired. The
underlying issue is use and misuse of the metaphorical
picture.
VI. FUTURE DIRECTIONS
We feel that further research on the role of language in
learning physics needs to examine more carefully the in-
structional implications of language as a legitimate repre-
sentation of knowledge and ideas in physics. For example,
how can we make students more aware of the presence
of physical models encoded in the metaphorical language
that we use? Can students be encouraged to think about
the applicability and limitations of different metaphor-
ical pictures [40]? Instead of allowing students to say,
“the electron is trapped in a square well,” unchallenged,
maybe the most important question to ask students is,
“what do you mean, what is this ‘square well’ you are
talking about?” As a corollary, maybe we should en-
courage students to ask us, “what do you mean?” when
we use a metaphor such as the electron is a wave
without justifying why it is applicable and when it is not.
Does it matter how we ask questions of our students?
If we phrase a question with different grammar or differ-
ent metaphors, do students respond differently? There
maybe occasions when the way in which the question is
asked is obscuring the real physical understanding that
students have. There is some preliminary evidence that
this may indeed be the case [54].
It seems to us that if we think of language as a repre-
sentation and recognize its unique difficulties, we should
put more effort into helping students become comfort-
able with this representation. Future research could fo-
cus on student difficulties in other areas of physics that
may be related to the language that students hear in the
physics classroom. (See [13] for example.) In some cases
difficulties may be related to linguistic models that stu-
dents have developed prior to instruction. We suggest
that some student difficulties may be attempts to nego-
tiate the meaning and applicability of different linguistic
models. Awareness of such linguistic difficulties would
help teachers to facilitate their students’ learning. More
research on this idea is needed.
There is one major aspect of cognitive linguistics that
we have not attempted to apply to the field physics edu-
cation research in this paper. This is the idea of concep-
tual blending [31]. Conceptual blending may provide a
complementary account of many of the ideas in this pa-
per. Conceptual blending also has an added advantage
in that it could account for online meaning construction
in terms of the blending of metaphors. This may bet-
ter account for “local” or “personal” ways of expression
observed among individual professors and students. The
dynamics of blending may also be useful for answering
questions about how we can make students more aware
of the myriad of models encoded by the metaphors in
physicists’ language. We think that this may be a fruit-
ful line of inquiry in future work.
Acknowledgments
We would like to thank the following people for their
help with this paper: L. Atkins, H. Brookes, G. Horton,
Y. Lin, J. Mestre, E. Redish, M. Sindel, A. Van Heuvelen,
A. Zech.
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