**Matrix Mechanics**

The theoretical ideas formulated in the seventeenth century by Isaac
Newton and Galileo reigned as the fundamental scientific precepts until
the year 1900, when Max Planck's work on the emission of light from a
hole in a heated hollow sphere showed that something was fundamentally
amiss. Planck's work identified a new constant of nature, called
Planck's quantum of action, that was alien to classical physics, and
that evidently needed to be integrated into a revised physics, to be
called quantum mechanics. A big step toward this new physics seemed to
be model of the atom devised in 1913 by Niels Bohr. It was a space-time
picture of the atom in which the electrons instead of spiraling inward
and gradually radiating away their energies, as demanded by classical
physics, were usually confined to stable orbits, which were specified in
terms of Planck's quantum of action.

The very strange thing about this model was that no light was emitted by
the circling electron when it was in one of these orbits. Light was
emitted, instead, when an electron jumped from one orbit to another.
However, its frequency was not some average of the frequencies of the
light that classical physics predicted should be emitted from the
electron of each of the two orbits: it was, instead, the *difference* of
these two frequencies.

A large amount of experimental data was being collected at that time
about the energy levels of various atoms, and about the rates at which
the transitions between different levels occurred. The excitations of
atoms from various states to more excited states could be induced by the
absorption of light, and the theory of this absorption and re-emission
of light was called dispersion theory.

Intensive efforts to construct a rationally coherent quantum mechanics
were being pursued by many groups, including most prominently those led
by Niels Bohr in Copenhagen, Max Born in Goettingen, and Arnold
Sommerfeld in Munich. But the key breakthrough was made by Werner
Heisenberg.

Heisenberg was a prodigy. He entered the University of Munich in 1920 at
age 18, and received his Ph. D three years later. In 1921 he published
with Sommerfeld's approval a bold and original paper on the anamolous
Zeeman effect, and in 1922 had co-authored two papers with Sommerfeld,
and had closely collaborated on another with Max Born. In September of
1924 he began a stay in Copenhagen where he collaborated with Bohr and
co-authored a paper on dispersion theory with Bohr's assistant Hendrik
Kramers. Thus when he returned to Goettingen in April of 1925 he was
only 23, but had spent the better part of five years working intensively
in close collaboration with the leaders of the field.

The state of affairs was at that point extremely muddled, with the
Copenhagen-based Bohr-Kramers-Slater dispersion theory recently
falsified by data. Also, a recent closely reasoned paper by Heisenberg's
close colleague, Wolfgang Pauli, argued that the entire program of
basing the theory on space-time pictures akin to Bohr's model was a
"swindle", and called for a new mathematical foundation: "It seems to
me...without doubt that not only the dynamical concept of force, but
also the kinematic concept of motion of classical theory, will have to
experience profound modification. ...I believe that the energy and
momentum values of the stationary states are much more real than
"orbits". (Pauli to Bohr, 12 December 1924).

Armed with all this deep knowledge and wise council, and influenced by
Einstein's 1905 success in shedding unhelpful intuitions and biases
concerning space and time by focusing on observable properties,
Heisenberg tried to find a new foundation for atomic physics based not
on a space-time picture of what was going on, but rather on mathematical
connections between observable quantities. The observables in the
abundant and accurate data pertaining to the dispersion of light were
energy levels of the "stationary states", whatever they were, and
transition amplitudes between these states. The transition amplitudes
refer to *two* states and thus form a square array. In order to
establish some sort of correspondence with the classical idea of an atom
Heisenberg needed arrays corresponding to the variable of classical
physics, such as momentum, position, acceleration, etc. and needed to
form the analogs of products of these "quantities". He constructed what
seemed to be the needed rules, by comparing to some apparently valid
rules of dispersion theory, and discovered that, for certain quantities
X and Y, XY was different from YX. This troubled Heisenberg, but did not
deter him.

Because atomic systems are complicated, Heisenberg considered first a
one- dimensional anharmonic oscillator, obtained by adding an extra
force term.

The results for that case, and in particular his proof that energy was
strictly conserved\-\--it was a violation of strict energy conservation
that had doomed the Bohr-Kramers-Slater theory\-\--convinced him that he
had found the basic structure he needed. Its subsequent successful
applications to innumerable physical situations by thousands of
physicists, with no proven failures. has borne out his optimism.

Born was quick to point out that the arrays of numbers, with their rule
of multiplication, were objects already well studied by mathematicians.
They are called "matrices", and the quantum theory based on them was,
for a time, called "matrix mechanics", particularly to distinguish it
from what appeared at first to be an alternative quantum mechanics
devised by Erwin Schroedinger, and called "wave mechanics". The two
theories were eventually shown to be formally equivalent by
Schroedinger, whose approach did seem to provide a space-time
description of the kind that Heisenberg and Pauli had deemed impossible.
However, Heisenberg, Pauli, and Bohr held that the Schroedinger wave was
an abstract formal structure that could be used to compute observable
quantities, because of the proved formal equivalence, but that it could
not be regarded as describing an actually existing space-time structure,
because of the "quantum jumps" that the wave needs to undergo in order
to keep it in line with human experience.

Because of the formal equivalence of the two forms, the two names
"matrix mechanics" and "wave mechanics" have largely fallen out of use
now, being replaced by the more inclusive names "quantum mechanics" and
"quantum theory".

The see how these ideas work in actual practice one may consider the
simplest case, in which the quantum system being examined has just two
states, labeled by an index i that can take two alternative possible
values, 1 or 2. Then the relevant arrays are sets of four (complex)
numbers z~ij~ where the two indices i and j each can take,
independently, the value 1 or 2. If one has two such sets z~ij~ and
w~ij~ then an array called (zw)~ij~ is defined by the rule (zw)~ij~ =
z~i1~w ~1j~ + z~i2~ w~2j~ .This is the standard rule of matrix
multiplication, in this two dimensional case.

Pauli defined four 2-by-2 matrices of interest:

(σ~0~) defined by ((σ~0~)~11~ = 1, (σ~0~)~12~ = 0, (σ~0~)~21~ = 0,
(σ~0~)~22~ =1),

(σ~1~) defined by ((σ~1~)~11~ = 0, (σ~1~)~12~ = 1, (σ~1~)~21~ = 1,
(σ~1~)~22~ = 0),

(σ~2~) defined by ((σ~2~)~11~ = 0, (σ~2~)~12~ = ─ i, (σ~2~)~21~ = i,
(σ~2~)~22~ = 0),

(σ~3~) defined by ((σ~3~)~11~ = 1, (σ~3~)~12~ = 0, (σ~3~)~21~ = 0,
(σ~3~)~22~ = ─ 1).

Laborious computations can then be simplified by writing matrices of
interest as linear combination: a= a~0~σ~0~ + a~1~σ~1~ + a~2~σ~2~ +
a~3~σ~3\ ,~ and using the following results:

for any i, σ~i~ σ ~i~ = σ~0;~ σ~o~σ~i~ = σ~i~σ~0~ = σ~i~ ; σ~1~σ~2~=
iσ~3~; σ~2~ σ~1~= ─ iσ~3~ .

These results follow directly from the definitions and multiplication
rules specified above. Notice that in the last two equations the order
in which the matrices are multiplied matters.

The rule that connects the mathematical symbols to our observations is
this:

Each elementary observation upon the system is associated with a
"projection operator" P. (Projection operators P must satisfy PP= P).

Let P~1~ be the projection operator that corresponds in the mathematics
to our knowledge that an associated set of preparation conditions have
been met.

Let P2 be the projection operator that corresponds in the mathematics to
the condition that a subsequent observation fulfills an associated set
of conditions. Then the predicted probability that a system known to be
prepared in accordance with the conditions corresponding to P1 at time
*t* = 0 will be observed at time *t* \> 0 to fulfill the conditions
corresponding to P2 is

Trace P~2~ (exp ─ iHt) P~1~ (exp iHt),

where H is the matrix that corresponds to energy, here assumed to have
no explicit dependence on time, and for any X, Trace X = X~11~ + X~22~,
for this 2-by-2 case. (I use units in which Planck's constant of action
is 2π.)

Suppose, for example, that P1 = (1+ σ~3~)/2, which corresponds the
prepared system's being in the state i = 1, and that P2 = (1 ─σ~3~)/2,
which corresponds to the system's being observed to be in the state i =
2. Suppose H= e σ~1.~

Using the fact (deducible from the power series expansion of exp x)

that (exp ─ ietσ~1~) = (cos et ─ i σ~1~ sin et) one can easily deduce
*just from the rules given above* that the probability identified above
is (sin et)^2^. The calculation is carried out without referring any
space-time picture of what is going on.

**Reference**

Mehra, J & Rechenberg, H: The Historical Development of Quantum
Mechanics, Volume 2. Chapters III, IV, & V.