# Special Relativity

## Relativistic Momentum

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate relativistic momentum.
2. Explain why the only mass it makes sense to talk about is rest mass.
In classical physics, momentum is a simple product of mass and velocity. However, we saw in the last section that when special relativity is taken into account, massive objects have a speed limit. What effect do you think mass and velocity have on the momentum of objects moving at relativistic speeds?
Momentum is one of the most important concepts in physics. The broadest form of Newton’s second law is stated in terms of momentum. Momentum is conserved whenever the net external force on a system is zero. This makes momentum conservation a fundamental tool for analyzing collisions. All of Work, Energy, and Energy Resources is devoted to momentum, and momentum has been important for many other topics as well, particularly where collisions were involved. We will see that momentum has the same importance in modern physics. Relativistic momentum is conserved, and much of what we know about subatomic structure comes from the analysis of collisions of accelerator-produced relativistic particles.
The first postulate of relativity states that the laws of physics are the same in all inertial frames. Does the law of conservation of momentum survive this requirement at high velocities? The answer is yes, provided that the momentum is defined as follows.
Note that we use  for velocity here to distinguish it from relative velocity  between observers. Only one observer is being considered here. With  defined in this way, total momentum  is conserved whenever the net external force is zero, just as in classical physics. Again we see that the relativistic quantity becomes virtually the same as the classical at low velocities. That is, relativistic momentum  becomes the classical  at low velocities, because  is very nearly equal to 1 at low velocities.
Relativistic momentum has the same intuitive feel as classical momentum. It is greatest for large masses moving at high velocities, but, because of the factor , relativistic momentum approaches infinity as  approaches . (See .) This is another indication that an object with mass cannot reach the speed of light. If it did, its momentum would become infinite, an unreasonable value.
Relativistic momentum is defined in such a way that the conservation of momentum will hold in all inertial frames. Whenever the net external force on a system is zero, relativistic momentum is conserved, just as is the case for classical momentum. This has been verified in numerous experiments.
In Relativistic Energy, the relationship of relativistic momentum to energy is explored. That subject will produce our first inkling that objects without mass may also have momentum.

### Section Summary

1. The law of conservation of momentum is valid whenever the net external force is zero and for relativistic momentum. Relativistic momentum  is classical momentum multiplied by the relativistic factor .
2. , where  is the rest mass of the object,  is its velocity relative to an observer, and the relativistic factor .
3. At low velocities, relativistic momentum is equivalent to classical momentum.
4. Relativistic momentum approaches infinity as  approaches . This implies that an object with mass cannot reach the speed of light.
5. Relativistic momentum is conserved, just as classical momentum is conserved.

### Conceptual Questions

### Problem Exercises


# Special Relativity

## Relativistic Energy

### Learning Objectives

By the end of this section, you will be able to:
1. Compute total energy of a relativistic object.
2. Compute the kinetic energy of a relativistic object.
3. Describe rest energy, and explain how it can be converted to other forms.
4. Explain why massive particles cannot reach C.
A tokamak is a form of experimental fusion reactor, which can change mass to energy. Accomplishing this requires an understanding of relativistic energy. Nuclear reactors are proof of the conservation of relativistic energy.
Conservation of energy is one of the most important laws in physics. Not only does energy have many important forms, but each form can be converted to any other. We know that classically the total amount of energy in a system remains constant. Relativistically, energy is still conserved, provided its definition is altered to include the possibility of mass changing to energy, as in the reactions that occur within a nuclear reactor. Relativistic energy is intentionally defined so that it will be conserved in all inertial frames, just as is the case for relativistic momentum. As a consequence, we learn that several fundamental quantities are related in ways not known in classical physics. All of these relationships are verified by experiment and have fundamental consequences. The altered definition of energy contains some of the most fundamental and spectacular new insights into nature found in recent history.

### Total Energy and Rest Energy

The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy is valid relativistically, if we define energy to include a relativistic factor.
This is the correct form of Einstein’s most famous equation, which for the first time showed that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. The implications of these first two equations regarding relativistic energy are so broad that they were not completely recognized for some years after Einstein published them in 1907, nor was the experimental proof that they are correct widely recognized at first. Einstein, it should be noted, did understand and describe the meanings and implications of his theory.
Today, the practical applications of the conversion of mass into another form of energy, such as in nuclear weapons and nuclear power plants, are well known. But examples also existed when Einstein first proposed the correct form of relativistic energy, and he did describe some of them. Nuclear radiation had been discovered in the previous decade, and it had been a mystery as to where its energy originated. The explanation was that, in certain nuclear processes, a small amount of mass is destroyed and energy is released and carried by nuclear radiation. But the amount of mass destroyed is so small that it is difficult to detect that any is missing. Although Einstein proposed this as the source of energy in the radioactive salts then being studied, it was many years before there was broad recognition that mass could be and, in fact, commonly is converted to energy. (See .)
Because of the relationship of rest energy to mass, we now consider mass to be a form of energy rather than something separate. There had not even been a hint of this prior to Einstein’s work. Such conversion is now known to be the source of the Sun’s energy, the energy of nuclear decay, and even the source of energy keeping Earth’s interior hot.

### Stored Energy and Potential Energy

What happens to energy stored in an object at rest, such as the energy put into a battery by charging it, or the energy stored in a toy gun’s compressed spring? The energy input becomes part of the total energy of the object and, thus, increases its rest mass. All stored and potential energy becomes mass in a system. Why is it we don’t ordinarily notice this? In fact, conservation of mass (meaning total mass is constant) was one of the great laws verified by 19th-century science. Why was it not noticed to be incorrect? The following example helps answer these questions.

### Kinetic Energy and the Ultimate Speed Limit

Kinetic energy is energy of motion. Classically, kinetic energy has the familiar expression . The relativistic expression for kinetic energy is obtained from the work-energy theorem. This theorem states that the net work on a system goes into kinetic energy. If our system starts from rest, then the work-energy theorem is
Relativistically, at rest we have rest energy . The work increases this to the total energy . Thus,
Relativistically, we have .
When motionless, we have  and
so that  at rest, as expected. But the expression for relativistic kinetic energy (such as total energy and rest energy) does not look much like the classical . To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for  at low velocities gives
A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In some cases, as in the limit of small velocity here, most terms are very small. Thus the expression derived for  here is not exact, but it is a very accurate approximation. Thus, at low velocities,
Entering this into the expression for relativistic kinetic energy gives
So, in fact, relativistic kinetic energy does become the same as classical kinetic energy when .
It is even more interesting to investigate what happens to kinetic energy when the velocity of an object approaches the speed of light. We know that  becomes infinite as  approaches , so that KErel also becomes infinite as the velocity approaches the speed of light. (See .) An infinite amount of work (and, hence, an infinite amount of energy input) is required to accelerate a mass to the speed of light.
So the speed of light is the ultimate speed limit for any particle having mass. All of this is consistent with the fact that velocities less than  always add to less than . Both the relativistic form for kinetic energy and the ultimate speed limit being  have been confirmed in detail in numerous experiments. No matter how much energy is put into accelerating a mass, its velocity can only approach—not reach—the speed of light.

### Relativistic Energy and Momentum

We know classically that kinetic energy and momentum are related to each other, since
Relativistically, we can obtain a relationship between energy and momentum by algebraically manipulating their definitions. This produces
where  is the relativistic total energy and  is the relativistic momentum. This relationship between relativistic energy and relativistic momentum is more complicated than the classical, but we can gain some interesting new insights by examining it. First, total energy is related to momentum and rest mass. At rest, momentum is zero, and the equation gives the total energy to be the rest energy  (so this equation is consistent with the discussion of rest energy above). However, as the mass is accelerated, its momentum  increases, thus increasing the total energy. At sufficiently high velocities, the rest energy term  becomes negligible compared with the momentum term

; thus,

 at extremely relativistic velocities.
If we consider momentum  to be distinct from mass, we can determine the implications of the equation   for a particle that has no mass. If we take  to be zero in this equation, then , or . Massless particles have this momentum. There are several massless particles found in nature, including photons (these are quanta of electromagnetic radiation). Another implication is that a massless particle must travel at speed  and only at speed . While it is beyond the scope of this text to examine the relationship in the equation  in detail, we can see that the relationship has important implications in special relativity.

### Test Prep for AP Courses

### Section Summary

1. Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
2. Total Energy is defined as:
, where

.
3. Rest energy is
,  meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy.
4. We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy.
5. The relativistic work-energy theorem is .
6. Relativistically,
 ,
where   is the relativistic kinetic energy.
7. Relativistic kinetic energy is , where . At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
8. No object with mass can attain the speed of light because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
9. The equation  relates the relativistic total energy

 and the relativistic momentum
. At extremely high velocities, the rest energy
 becomes negligible, and
.

### Conceptual Questions

### Problems & Exercises


# Quantum Physics

## Connection for AP® Courses

In this chapter, the basic principles of quantum mechanics are introduced. Quantum mechanics is the branch of physics needed to deal with submicroscopic objects. Because these objects are smaller than those, such as computers, books, or cars, that we can observe directly with our senses, and so generally must be observed with the aid of instruments, parts of quantum mechanics seem as foreign and bizarre as the effects of relative motion near the speed of light. Yet through experimental results, quantum mechanics has been shown to be valid. Truth is often stranger than fiction.
Quantum theory was developed initially to explain the behavior of electromagnetic energy in certain situations, such as blackbody radiation or the photoelectric effect, which could not be understood in terms of classical electrodynamics (Essential Knowledge 1.D.2). In the quantum model, light is treated as a packet of energy called a photon, which has both the properties of a wave and a particle (Essential Knowledge 6.F.3). The energy of a photon is directly proportional to its frequency.
This new model for light provided the foundation for one of the most important ideas in quantum theory: wave-particle duality. Just as light has properties of both waves and particles, matter also has the properties of waves and particles (Essential Knowledge 1.D.1). This interpretation of matter and energy explained observations at the atomic level that could not be explained by classical mechanics or electromagnetic theory (Enduring Understanding 1.D). The quantum interpretation of energy and matter at the atomic level, most notably the internal structure of atoms, supports Big Idea 1 of the AP Physics Curriculum Framework.
Big Idea 1 is also supported by the correspondence principle. Classical mechanics cannot accurately describe systems at the atomic level, whereas quantum mechanics is able to describe systems at both levels. However, the properties of matter that are described by waves become insignificant at the macroscopic level, so that for large systems of matter, the quantum description closely approaches, or corresponds to, the classical description (Essential Knowledge 6.G.1, Essential Knowledge 6.G.2, Essential Knowledge 6.F.3).
Big Ideas 5 and 6 are supported by the descriptions of energy and momentum transfer at the quantum level. Although quantum mechanics overturned a number of fundamental ideas of classical physics, the most important principles, such as energy conservation and momentum conservation, remained intact (Enduring Understanding 5.B, Enduring Understanding 5.D). Quantum mechanics expands on these principles, so that the particle-like behavior of electromagnetic energy describes momentum transfer, while the wave-like behavior of matter accounts for why electrons produce diffraction patterns when they pass through the atomic lattices of crystals.
At the quantum level, the effects of measurement are very different from those at the macroscopic level. Because the wave properties of matter are more prominent for small particles, such as electrons, and a wave does not have a specific location, the position and momentum of matter cannot be measured with absolute precision (Essential Knowledge 1.D.3). Rather, the particle has a certain probability of being in a location interval for a specific momentum, or being located within a particular interval of time for a specific energy (Enduring Understanding 7.C, Essential Knowledge 7.C.1). These probabilistic limits on measurement are described by Heisenberg’s uncertainty principle, which connects wave-particle duality to the non-absolute properties of space and time. At the quantum level, measurements affect the system being measured, and so restrict the degree to which properties can be known. The discussion of this probabilistic interpretation supports Big Idea 7 of the AP Physics Curriculum Framework.
The concepts in this chapter support:
Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure.
Enduring Understanding 1.D Classical mechanics cannot describe all properties of objects.
Essential Knowledge 1.D.1 Objects classically thought of as particles can exhibit properties of waves.
Essential Knowledge 1.D.2 Certain phenomena classically thought of as waves can exhibit properties of particles.
Essential Knowledge 1.D.3 Properties of space and time cannot always be treated as absolute.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.8 Energy transfer occurs when photons are absorbed or emitted, for example, by atoms or nuclei.
Enduring Understanding 5.D The linear momentum of a system is conserved.
Essential Knowledge 5.D.1 In a collision between objects, linear momentum is conserved. In an elastic collision, kinetic energy is the same before and after.
Big Idea 6 Waves can transfer energy and momentum from one location to another without the permanent transfer of mass and serve as a mathematical model for the description of other phenomena.
Enduring Understanding 6.F Electromagnetic radiation can be modeled as waves or as fundamental particles.
Essential Knowledge 6.F.3 Photons are individual energy packets of electromagnetic waves, with Ephoton = hf, where h is Planck’s constant and f is the frequency of the associated light wave.
Essential Knowledge 6.F.4 The nature of light requires that different models of light are most appropriate at different scales.
Enduring Understanding 6.G All matter can be modeled as waves or as particles.
Essential Knowledge 6.G.1 Under certain regimes of energy or distance, matter can be modeled as a classical particle.
Essential Knowledge 6.G.2 Under certain regimes of energy or distance, matter can be modeled as a wave. The behavior in these regimes is described by quantum mechanics.
Big Idea 7. The mathematics of probability can be used to describe the behavior of complex systems and to interpret the behavior of quantum mechanical systems.
Enduring Understanding 7.C At the quantum scale, matter is described by a wave function, which leads to a probabilistic description of the microscopic world.
Essential Knowledge 7.C.1 The probabilistic description of matter is modeled by a wave function, which can be assigned to an object and used to describe its motion and interactions. The absolute value of the wave function is related to the probability of ﬁnding a particle in some spatial region. (Qualitative treatment only, using graphical analysis.)
Atoms, molecules, and fundamental electron and proton charges are all examples of physical entities that are quantized—that is, they appear only in certain discrete values and do not have every conceivable value. Quantized is the opposite of continuous. We cannot have a fraction of an atom, or part of an electron’s charge, or 14-1/3 cents, for example. Rather, everything is built of integral multiples of these substructures. Quantum physics is the branch of physics that deals with small objects and the quantization of various entities, including energy and angular momentum. Just as with classical physics, quantum physics has several subfields, such as mechanics and the study of electromagnetic forces. The correspondence principle states that in the classical limit (large, slow-moving objects), quantum mechanics becomes the same as classical physics. In this chapter, we begin the development of quantum mechanics and its description of the strange submicroscopic world. In later chapters, we will examine many areas, such as atomic and nuclear physics, in which quantum mechanics is crucial.


# Quantum Physics

## Quantization of Energy

### Learning Objectives

By the end of this section, you will be able to:
1. Explain Max Planck’s contribution to the development of quantum mechanics.
2. Explain why atomic spectra indicate quantization.

### Planck’s Contribution

Energy is quantized in some systems, meaning that the system can have only certain energies and not a continuum of energies, unlike the classical case. This would be like having only certain speeds at which a car can travel because its kinetic energy can have only certain values. We also find that some forms of energy transfer take place with discrete lumps of energy. While most of us are familiar with the quantization of matter into lumps called atoms, molecules, and the like, we are less aware that energy, too, can be quantized. Some of the earliest clues about the necessity of quantum mechanics over classical physics came from the quantization of energy.
Where is the quantization of energy observed? Let us begin by considering the emission and absorption of electromagnetic (EM) radiation. The EM spectrum radiated by a hot solid is linked directly to the solid’s temperature. (See .) An ideal radiator is one that has an emissivity of 1 at all wavelengths and, thus, is jet black. Ideal radiators are therefore called blackbodies, and their EM radiation is called blackbody radiation. It was discussed that the total intensity of the radiation varies as  the fourth power of the absolute temperature of the body, and that the peak of the spectrum shifts to shorter wavelengths at higher temperatures. All of this seems quite continuous, but it was the curve of the spectrum of intensity versus wavelength that gave a clue that the energies of the atoms in the solid are quantized. In fact, providing a theoretical explanation for the experimentally measured shape of the spectrum was a mystery at the turn of the century. When this “ultraviolet catastrophe” was eventually solved, the answers led to new technologies such as computers and the sophisticated imaging techniques described in earlier chapters. Once again, physics as an enabling science changed the way we live.
The German physicist Max Planck (1858–1947) used the idea that atoms and molecules in a body act like oscillators to absorb and emit radiation. The energies of the oscillating atoms and molecules had to be quantized to correctly describe the shape of the blackbody spectrum. Planck deduced that the energy of an oscillator having a frequency  is given by
Here  is any nonnegative integer (0, 1, 2, 3, …). The symbol  stands for Planck’s constant, given by
The equation  means that an oscillator having a frequency  (emitting and absorbing EM radiation of frequency ) can have its energy increase or decrease only in discrete steps of size
It might be helpful to mention some macroscopic analogies of this quantization of energy phenomena. This is like a pendulum that has a characteristic oscillation frequency but can swing with only certain amplitudes. Quantization of energy also resembles a standing wave on a string that allows only particular harmonics described by integers. It is also similar to going up and down a hill using discrete stair steps rather than being able to move up and down a continuous slope. Your potential energy takes on discrete values as you move from step to step.
Using the quantization of oscillators, Planck was able to correctly describe the experimentally known shape of the blackbody spectrum. This was the first indication that energy is sometimes quantized on a small scale and earned him the Nobel Prize in Physics in 1918. Although Planck’s theory comes from observations of a macroscopic object, its analysis is based on atoms and molecules. It was such a revolutionary departure from classical physics that Planck himself was reluctant to accept his own idea that energy states are not continuous. The general acceptance of Planck’s energy quantization was greatly enhanced by Einstein’s explanation of the photoelectric effect (discussed in the next section), which took energy quantization a step further. Planck was fully involved in the development of both early quantum mechanics and relativity. He quickly embraced Einstein’s special relativity, published in 1905, and in 1906 Planck was the first to suggest the correct formula for relativistic momentum, .
Note that Planck’s constant  is a very small number. So for an infrared frequency of  being emitted by a blackbody, for example, the difference between energy levels is only  or about 0.4 eV. This 0.4 eV of energy is significant compared with typical atomic energies, which are on the order of an electron volt, or thermal energies, which are typically fractions of an electron volt. But on a macroscopic or classical scale, energies are typically on the order of joules. Even if macroscopic energies are quantized, the quantum steps are too small to be noticed. This is an example of the correspondence principle. For a large object, quantum mechanics produces results indistinguishable from those of classical physics.

### Atomic Spectra

Now let us turn our attention to the emission and absorption of EM radiation by gases. The Sun is the most common example of a body containing gases emitting an EM spectrum that includes visible light. We also see examples in neon signs and candle flames. Studies of emissions of hot gases began more than two centuries ago, and it was soon recognized that these emission spectra contained huge amounts of information. The type of gas and its temperature, for example, could be determined. We now know that these EM emissions come from electrons transitioning between energy levels in individual atoms and molecules; thus, they are called atomic spectra. Atomic spectra remain an important analytical tool today.  shows an example of an emission spectrum obtained by passing an electric discharge through a material. One of the most important characteristics of these spectra is that they are discrete. By this we mean that only certain wavelengths, and hence frequencies, are emitted. This is called a line spectrum. If frequency and energy are associated as  the energies of the electrons in the emitting atoms and molecules are quantized. This is discussed in more detail later in this chapter.
It was a major puzzle that atomic spectra are quantized. Some of the best minds of 19th-century science failed to explain why this might be. Not until the second decade of the 20th century did an answer based on quantum mechanics begin to emerge. Again a macroscopic or classical body of gas was involved in the studies, but the effect, as we shall see, is due to individual atoms and molecules.

### Test Prep for AP Courses

### Section Summary

1. The first indication that energy is sometimes quantized came from blackbody radiation, which is the emission of EM radiation by an object with an emissivity of 1.
2. Planck recognized that the energy levels of the emitting atoms and molecules were quantized, with only the allowed values of  where  is any non-negative integer (0, 1, 2, 3, …).
3.  is Planck’s constant, whose value is
4. Thus, the oscillatory absorption and emission energies of atoms and molecules in a blackbody could increase or decrease only in steps of size  where  is the frequency of the oscillatory nature of the absorption and emission of EM radiation.
5. Another indication of energy levels being quantized in atoms and molecules comes from the lines in atomic spectra, which are the EM emissions of individual atoms and molecules.

### Conceptual Questions

### Problems & Exercises


# Quantum Physics

## The Photoelectric Effect

### Learning Objectives

By the end of this section, you will be able to:
1. Describe a typical photoelectric-effect experiment.
2. Determine the maximum kinetic energy of photoelectrons ejected by photons of one energy or wavelength, when given the maximum kinetic energy of photoelectrons for a different photon energy or wavelength.
When light strikes materials, it can eject electrons from them. This is called the photoelectric effect, meaning that light (photo) produces electricity. One common use of the photoelectric effect is in light meters, such as those that adjust the automatic iris on various types of cameras. In a similar way, another use is in solar cells, as you probably have in your calculator or have seen on a roof top or a roadside sign. These make use of the photoelectric effect to convert light into electricity for running different devices.
This effect has been known for more than a century and can be studied using a device such as that shown in . This figure shows an evacuated tube with a metal plate and a collector wire that are connected by a variable voltage source, with the collector more negative than the plate. When light (or other EM radiation) strikes the plate in the evacuated tube, it may eject electrons. If the electrons have energy in electron volts (eV) greater than the potential difference between the plate and the wire in volts, some electrons will be collected on the wire. Since the electron energy in eV is , where  is the electron charge and  is the potential difference, the electron energy can be measured by adjusting the retarding voltage between the wire and the plate. The voltage that stops the electrons from reaching the wire equals the energy in eV. For example, if  barely stops the electrons, their energy is 3.00 eV. The number of electrons ejected can be determined by measuring the current between the wire and plate. The more light, the more electrons; a little circuitry allows this device to be used as a light meter.
What is really important about the photoelectric effect is what Albert Einstein deduced from it. Einstein realized that there were several characteristics of the photoelectric effect that could be explained only if EM radiation is itself quantized: the apparently continuous stream of energy in an EM wave is actually composed of energy quanta called photons. In his explanation of the photoelectric effect, Einstein defined a quantized unit or quantum of EM energy, which we now call a photon, with an energy proportional to the frequency of EM radiation. In equation form, the photon energy is
where  is the energy of a photon of frequency  and  is Planck’s constant. This revolutionary idea looks similar to Planck’s quantization of energy states in blackbody oscillators, but it is quite different. It is the quantization of EM radiation itself. EM waves are composed of photons and are not continuous smooth waves as described in previous chapters on optics. Their energy is absorbed and emitted in lumps, not continuously. This is exactly consistent with Planck’s quantization of energy levels in blackbody oscillators, since these oscillators increase and decrease their energy in steps of  by absorbing and emitting photons having . We do not observe this with our eyes, because there are so many photons in common light sources that individual photons go unnoticed. (See .) The next section of the text (Photon Energies and the Electromagnetic Spectrum) is devoted to a discussion of photons and some of their characteristics and implications. For now, we will use the photon concept to explain the photoelectric effect, much as Einstein did.
The photoelectric effect has the properties discussed below. All these properties are consistent with the idea that individual photons of EM radiation are absorbed by individual electrons in a material, with the electron gaining the photon’s energy. Some of these properties are inconsistent with the idea that EM radiation is a simple wave. For simplicity, let us consider what happens with monochromatic EM radiation in which all photons have the same energy .
1. If we vary the frequency of the EM radiation falling on a material, we find the following: For a given material, there is a threshold frequency  for the EM radiation below which no electrons are ejected, regardless of intensity. Individual photons interact with individual electrons. Thus if the photon energy is too small to break an electron away, no electrons will be ejected. If EM radiation was a simple wave, sufficient energy could be obtained by increasing the intensity.
2. Once EM radiation falls on a material, electrons are ejected without delay. As soon as an individual photon of a sufficiently high frequency is absorbed by an individual electron, the electron is ejected. If the EM radiation were a simple wave, several minutes would be required for sufficient energy to be deposited to the metal surface to eject an electron.
3. The number of electrons ejected per unit time is proportional to the intensity of the EM radiation and to no other characteristic. High-intensity EM radiation consists of large numbers of photons per unit area, with all photons having the same characteristic energy .
4. If we vary the intensity of the EM radiation and measure the energy of ejected electrons, we find the following: The maximum kinetic energy of ejected electrons is independent of the intensity of the EM radiation. Since there are so many electrons in a material, it is extremely unlikely that two photons will interact with the same electron at the same time, thereby increasing the energy given it. Instead (as noted in 3 above), increased intensity results in more electrons of the same energy being ejected. If EM radiation were a simple wave, a higher intensity could give more energy, and higher-energy electrons would be ejected.
5. The kinetic energy of an ejected electron equals the photon energy minus the binding energy of the electron in the specific material. An individual photon can give all of its energy to an electron. The photon’s energy is partly used to break the electron away from the material. The remainder goes into the ejected electron’s kinetic energy. In equation form, this is given by
 where  is the maximum kinetic energy of the ejected electron,  is the photon’s energy, and BE is the binding energy of the electron to the particular material. (BE is sometimes called the work function of the material.) This equation, due to Einstein in 1905, explains the properties of the photoelectric effect quantitatively. An individual photon of EM radiation (it does not come any other way) interacts with an individual electron, supplying enough energy, BE, to break it away, with the remainder going to kinetic energy. The binding energy is , where   is the threshold frequency for the particular material.  shows a graph of maximum  versus the frequency of incident EM radiation falling on a particular material.
Einstein’s idea that EM radiation is quantized was crucial to the beginnings of quantum mechanics. It is a far more general concept than its explanation of the photoelectric effect might imply. All EM radiation can also be modeled in the form of photons, and the characteristics of EM radiation are entirely consistent with this fact. (As we will see in the next section, many aspects of EM radiation, such as the hazards of ultraviolet (UV) radiation, can be explained only by photon properties.) More famous for modern relativity, Einstein planted an important seed for quantum mechanics in 1905, the same year he published his first paper on special relativity. His explanation of the photoelectric effect was the basis for the Nobel Prize awarded to him in 1921. Although his other contributions to theoretical physics were also noted in that award, special and general relativity were not fully recognized in spite of having been partially verified by experiment by 1921. Although hero-worshipped, this great man never received Nobel recognition for his most famous work—relativity.

### Test Prep for AP Courses

### Section Summary

1. The photoelectric effect is the process in which EM radiation ejects electrons from a material.
2. Einstein proposed photons to be quanta of EM radiation having energy , where  is the frequency of the radiation.
3. All EM radiation is composed of photons. As Einstein explained, all characteristics of the photoelectric effect are due to the interaction of individual photons with individual electrons.
4. The maximum kinetic energy  of ejected electrons (photoelectrons) is given by , where  is the photon energy and BE is the binding energy (or work function) of the electron to the particular material.

### Conceptual Questions

### Problems & Exercises


# Quantum Physics

## Photon Energies and the Electromagnetic Spectrum

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the relationship between the energy of a photon in joules or electron volts and its wavelength or frequency.
2. Calculate the number of photons per second emitted by a monochromatic source of specific wavelength and power.

### Ionizing Radiation

A photon is a quantum of EM radiation. Its energy is given by  and is related to the frequency  and wavelength  of the radiation by
where  is the energy of a single photon and  is the speed of light. When working with small systems, energy in eV is often useful. Note that Planck’s constant in these units is
Since many wavelengths are stated in nanometers (nm), it is also useful to know that
These will make many calculations a little easier.
All EM radiation is composed of photons.  shows various divisions of the EM spectrum plotted against wavelength, frequency, and photon energy. Previously in this book, photon characteristics were alluded to in the discussion of some of the characteristics of UV, x rays, and  rays, the first of which start with frequencies just above violet in the visible spectrum. It was noted that these types of EM radiation have characteristics much different than visible light. We can now see that such properties arise because photon energy is larger at high frequencies.
Photons act as individual quanta and interact with individual electrons, atoms, molecules, and so on. The energy a photon carries is, thus, crucial to the effects it has.  lists representative submicroscopic energies in eV. When we compare photon energies from the EM spectrum in  with energies in the table, we can see how effects vary with the type of EM radiation.
Gamma rays, a form of nuclear and cosmic EM radiation, can have the highest frequencies and, hence, the highest photon energies in the EM spectrum. For example, a -ray photon with  has an energy  This is sufficient energy to ionize thousands of atoms and molecules, since only 10 to 1000 eV are needed per ionization. In fact,  rays are one type of ionizing radiation, as are x rays and UV, because they produce ionization in materials that absorb them. Because so much ionization can be produced, a single -ray photon can cause significant damage to biological tissue, killing cells or damaging their ability to properly reproduce. When cell reproduction is disrupted, the result can be cancer, one of the known effects of exposure to ionizing radiation. Since cancer cells are rapidly reproducing, they are exceptionally sensitive to the disruption produced by ionizing radiation. This means that ionizing radiation has positive uses in cancer treatment as well as risks in producing cancer.
High photon energy also enables  rays to penetrate materials, since a collision with a single atom or molecule is unlikely to absorb all the  ray’s energy. This can make  rays useful as a probe, and they are sometimes used in medical imaging. x rays, as you can see in , overlap with the low-frequency end of the  ray range. Since x rays have energies of keV and up, individual x-ray photons also can produce large amounts of ionization. At lower photon energies, x rays are not as penetrating as  rays and are slightly less hazardous. X rays are ideal for medical imaging, their most common use, and a fact that was recognized immediately upon their discovery in 1895 by the German physicist W. C. Roentgen (1845–1923). (See .) Within one year of their discovery, x rays (for a time called Roentgen rays) were used for medical diagnostics. Roentgen received the 1901 Nobel Prize for the discovery of x rays.
While  rays originate in nuclear decay, x rays are produced by the process shown in . Electrons ejected by thermal agitation from a hot filament in a vacuum tube are accelerated through a high voltage, gaining kinetic energy from the electrical potential energy. When they strike the anode, the electrons convert their kinetic energy to a variety of forms, including thermal energy. But since an accelerated charge radiates EM waves, and since the electrons act individually, photons are also produced. Some of these x-ray photons obtain the kinetic energy of the electron. The accelerated electrons originate at the cathode, so such a tube is called a cathode ray tube (CRT), and various versions of them are found in older TV and computer screens as well as in x-ray machines.
 shows the spectrum of x rays obtained from an x-ray tube. There are two distinct features to the spectrum. First, the smooth distribution results from electrons being decelerated in the anode material. A curve like this is obtained by detecting many photons, and it is apparent that the maximum energy is unlikely. This decelerating process produces radiation that is called bremsstrahlung (German for braking radiation). The second feature is the existence of sharp peaks in the spectrum; these are called characteristic x rays, since they are characteristic of the anode material. Characteristic x rays come from atomic excitations unique to a given type of anode material. They are akin to lines in atomic spectra, implying the energy levels of atoms are quantized. Phenomena such as discrete atomic spectra and characteristic x rays are explored further in Atomic Physics.
Ultraviolet radiation (approximately 4 eV to 300 eV) overlaps with the low end of the energy range of x rays, but UV is typically lower in energy. UV comes from the de-excitation of atoms that may be part of a hot solid or gas. These atoms can be given energy that they later release as UV by numerous processes, including electric discharge, nuclear explosion, thermal agitation, and exposure to x rays. A UV photon has sufficient energy to ionize atoms and molecules, which makes its effects different from those of visible light. UV thus has some of the same biological effects as  rays and x rays. For example, it can cause skin cancer and is used as a sterilizer. The major difference is that several UV photons are required to disrupt cell reproduction or kill a bacterium, whereas single -ray and X-ray photons can do the same damage. But since UV does have the energy to alter molecules, it can do what visible light cannot. One of the beneficial aspects of UV is that it triggers the production of vitamin D in the skin, whereas visible light has insufficient energy per photon to alter the molecules that trigger this production. Infantile jaundice is treated by exposing the baby to UV (with eye protection), called phototherapy, the beneficial effects of which are thought to be related to its ability to help prevent the buildup of potentially toxic bilirubin in the blood.

### Visible Light

The range of photon energies for visible light from red to violet is 1.63 to 3.26 eV, respectively (left for this chapter’s Problems and Exercises to verify). These energies are on the order of those between outer electron shells in atoms and molecules. This means that these photons can be absorbed by atoms and molecules. A single photon can actually stimulate the retina, for example, by altering a receptor molecule that then triggers a nerve impulse. Photons can be absorbed or emitted only by atoms and molecules that have precisely the correct quantized energy step to do so. For example, if a red photon of frequency  encounters a molecule that has an energy step,  equal to  then the photon can be absorbed. Violet flowers absorb red and reflect violet; this implies there is no energy step between levels in the receptor molecule equal to the violet photon’s energy, but there is an energy step for the red.
There are some noticeable differences in the characteristics of light between the two ends of the visible spectrum that are due to photon energies. Red light has insufficient photon energy to expose most black-and-white film, and it is thus used to illuminate darkrooms where such film is developed. Since violet light has a higher photon energy, dyes that absorb violet tend to fade more quickly than those that do not. (See .) Take a look at some faded color posters in a storefront some time, and you will notice that the blues and violets are the last to fade. This is because other dyes, such as red and green dyes, absorb blue and violet photons, the higher energies of which break up their weakly bound molecules. (Complex molecules such as those in dyes and DNA tend to be weakly bound.) Blue and violet dyes reflect those colors and, therefore, do not absorb these more energetic photons, thus suffering less molecular damage.
Transparent materials, such as some glasses, do not absorb any visible light, because there is no energy step in the atoms or molecules that could absorb the light. Since individual photons interact with individual atoms, it is nearly impossible to have two photons absorbed simultaneously to reach a large energy step. Because of its lower photon energy, visible light can sometimes pass through many kilometers of a substance, while higher frequencies like UV, x ray, and  rays are absorbed, because they have sufficient photon energy to ionize the material.

### Lower-Energy Photons

Infrared radiation (IR) has even lower photon energies than visible light and cannot significantly alter atoms and molecules. IR can be absorbed and emitted by atoms and molecules, particularly between closely spaced states. IR is extremely strongly absorbed by water, for example, because water molecules have many states separated by energies on the order of  to  well within the IR and microwave energy ranges. This is why in the IR range, skin is almost jet black, with an emissivity near 1—there are many states in water molecules in the skin that can absorb a large range of IR photon energies. Not all molecules have this property. Air, for example, is nearly transparent to many IR frequencies.
Microwaves are the highest frequencies that can be produced by electronic circuits, although they are also produced naturally. Thus microwaves are similar to IR but do not extend to as high frequencies. There are states in water and other molecules that have the same frequency and energy as microwaves, typically about  This is one reason why food absorbs microwaves more strongly than many other materials, making microwave ovens an efficient way of putting energy directly into food.
Photon energies for both IR and microwaves are so low that huge numbers of photons are involved in any significant energy transfer by IR or microwaves (such as warming yourself with a heat lamp or cooking pizza in the microwave). Visible light, IR, microwaves, and all lower frequencies cannot produce ionization with single photons and do not ordinarily have the hazards of higher frequencies. When visible, IR, or microwave radiation is hazardous, such as the inducement of cataracts by microwaves, the hazard is due to huge numbers of photons acting together (not to an accumulation of photons, such as sterilization by weak UV). The negative effects of visible, IR, or microwave radiation can be thermal effects, which could be produced by any heat source. But one difference is that at very high intensity, strong electric and magnetic fields can be produced by photons acting together. Such electromagnetic fields (EMF) can actually ionize materials.
It is virtually impossible to detect individual photons having frequencies below microwave frequencies, because of their low photon energy. But the photons are there. A continuous EM wave can be modeled as photons. At low frequencies, EM waves are generally treated as time- and position-varying electric and magnetic fields with no discernible quantization. This is another example of the correspondence principle in situations involving huge numbers of photons.

### Test Prep for AP Courses

### Section Summary

1. Photon energy is responsible for many characteristics of EM radiation, being particularly noticeable at high frequencies.
2. Photons have both wave and particle characteristics.

### Conceptual Questions

### Problems & Exercises


# Quantum Physics

## Photon Momentum

### Learning Objectives

By the end of this section, you will be able to:
1. Relate the linear momentum of a photon to its energy or wavelength, and apply linear momentum conservation to simple processes involving the emission, absorption, or reflection of photons.
2. Account qualitatively for the increase of photon wavelength that is observed, and explain the significance of the Compton wavelength.

### Measuring Photon Momentum

The quantum of EM radiation we call a photon has properties analogous to those of particles we can see, such as grains of sand. A photon interacts as a unit in collisions or when absorbed, rather than as an extensive wave. Massive quanta, like electrons, also act like macroscopic particles—something we expect, because they are the smallest units of matter. Particles carry momentum as well as energy. Despite photons having no mass, there has long been evidence that EM radiation carries momentum. (Maxwell and others who studied EM waves predicted that they would carry momentum.) It is now a well-established fact that photons do have momentum. In fact, photon momentum is suggested by the photoelectric effect, where photons knock electrons out of a substance.  shows macroscopic evidence of photon momentum.
 shows a comet with two prominent tails. What most people do not know about the tails is that they always point away from the Sun rather than trailing behind the comet (like the tail of Bo Peep’s sheep). Comet tails are composed of gases and dust evaporated from the body of the comet and ionized gas. The dust particles recoil away from the Sun when photons scatter from them. Evidently, photons carry momentum in the direction of their motion (away from the Sun), and some of this momentum is transferred to dust particles in collisions. Gas atoms and molecules in the blue tail are most affected by other particles of radiation, such as protons and electrons emanating from the Sun, rather than by the momentum of photons.
Momentum is conserved in quantum mechanics just as it is in relativity and classical physics. Some of the earliest direct experimental evidence of this came from scattering of x-ray photons by electrons in substances, named Compton scattering after the American physicist Arthur H. Compton (1892–1962). Around 1923, Compton observed that x rays scattered from materials had a decreased energy and correctly analyzed this as being due to the scattering of photons from electrons. This phenomenon could be handled as a collision between two particles—a photon and an electron at rest in the material. Energy and momentum are conserved in the collision. (See ) He won a Nobel Prize in 1929 for the discovery of this scattering, now called the Compton effect, because it helped prove that photon momentum is given by
where  is Planck’s constant and  is the photon wavelength. (Note that relativistic momentum given as  is valid only for particles having mass.)
We can see that photon momentum is small, since  and  is very small. It is for this reason that we do not ordinarily observe photon momentum. Our mirrors do not recoil when light reflects from them (except perhaps in cartoons). Compton saw the effects of photon momentum because he was observing x rays, which have a small wavelength and a relatively large momentum, interacting with the lightest of particles, the electron.

### Relativistic Photon Momentum

There is a relationship between photon momentum  and photon energy  that is consistent with the relation given previously for the relativistic total energy of a particle as . We know  is zero for a photon, but  is not, so that

 becomes
or
To check the validity of this relation, note that  for a photon. Substituting this into  yields
as determined experimentally and discussed above. Thus,  is equivalent to Compton’s result . For a further verification of the relationship between photon energy and momentum, see .

### Test Prep for AP Courses

### Section Summary

1. Photons have momentum, given by , where  is the photon wavelength.
2. Photon energy and momentum are related by
 , where  for a photon.

### Conceptual Questions

### Problems & Exercises


# Quantum Physics

## The Particle-Wave Duality

### Learning Objectives

By the end of this section, you will be able to:
1. Explain what the term particle-wave duality means, and why it is applied to EM radiation.
We have long known that EM radiation is a wave, capable of interference and diffraction. We now see that light can be modeled as photons, which are massless particles. This may seem contradictory, since we ordinarily deal with large objects that never act like both wave and particle. An ocean wave, for example, looks nothing like a rock. To understand small-scale phenomena, we make analogies with the large-scale phenomena we observe directly. When we say something behaves like a wave, we mean it shows interference effects analogous to those seen in overlapping water waves. (See .) Two examples of waves are sound and EM radiation. When we say something behaves like a particle, we mean that it interacts as a discrete unit with no interference effects. Examples of particles include electrons, atoms, and photons of EM radiation. How do we talk about a phenomenon that acts like both a particle and a wave?
There is no doubt that EM radiation interferes and has the properties of wavelength and frequency. There is also no doubt that it behaves as particlesâ€”photons with discrete energy. We call this twofold nature the particle-wave duality, meaning that EM radiation has both particle and wave properties. This so-called duality is simply a term for properties of the photon analogous to phenomena we can observe directly, on a macroscopic scale. If this term seems strange, it is because we do not ordinarily observe details on the quantum level directly, and our observations yield either particle or wavelike properties, but never both simultaneously.
Since we have a particle-wave duality for photons, and since we have seen connections between photons and matter in that both have momentum, it is reasonable to ask whether there is a particle-wave duality for matter as well. If the EM radiation we once thought to be a pure wave has particle properties, is it possible that matter has wave properties? The answer is yes. The consequences are tremendous, as we will begin to see in the next section.

### Test Prep for AP Courses

### Section Summary

1. EM radiation can behave like either a particle or a wave.
2. This is termed particle-wave duality.


# Quantum Physics

## The Wave Nature of Matter

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the Davisson-Germer experiment, and explain how it provides evidence for the wave nature of electrons.

### De Broglie Wavelength

In 1923 a French physics graduate student named Prince Louis-Victor de Broglie (1892–1987) made a radical proposal based on the hope that nature is symmetric. If EM radiation has both particle and wave properties, then nature would be symmetric if matter also had both particle and wave properties. If what we once thought of as an unequivocal wave (EM radiation) is also a particle, then what we think of as an unequivocal particle (matter) may also be a wave. De Broglie’s suggestion, made as part of his doctoral thesis, was so radical that it was greeted with some skepticism. A copy of his thesis was sent to Einstein, who said it was not only probably correct, but that it might be of fundamental importance. With the support of Einstein and a few other prominent physicists, de Broglie was awarded his doctorate.
De Broglie took both relativity and quantum mechanics into account to develop the proposal that all particles have a wavelength, given by
where  is Planck’s constant and  is momentum. This is defined to be the de Broglie wavelength. (Note that we already have this for photons, from the equation .) The hallmark of a wave is interference. If matter is a wave, then it must exhibit constructive and destructive interference. Why isn’t this ordinarily observed? The answer is that in order to see significant interference effects, a wave must interact with an object about the same size as its wavelength. Since  is very small,  is also small, especially for macroscopic objects. A 3-kg bowling ball moving at 10 m/s, for example, has  This means that to see its wave characteristics, the bowling ball would have to interact with something about  in size—far smaller than anything known. When waves interact with objects much larger than their wavelength, they show negligible interference effects and move in straight lines (such as light rays in geometric optics). To get easily observed interference effects from particles of matter, the longest wavelength and hence smallest mass possible would be useful. Therefore, this effect was first observed with electrons.
American physicists Clinton J. Davisson and Lester H. Germer in 1925 and, independently, British physicist G. P. Thomson (son of J. J. Thomson, discoverer of the electron) in 1926 scattered electrons from crystals and found diffraction patterns. These patterns are exactly consistent with interference of electrons having the de Broglie wavelength and are somewhat analogous to light interacting with a diffraction grating. (See .)
De Broglie’s proposal of a wave nature for all particles initiated a remarkably productive era in which the foundations for quantum mechanics were laid. In 1926, the Austrian physicist Erwin Schrödinger (1887–1961) published four papers in which the wave nature of particles was treated explicitly with wave equations. At the same time, many others began important work. Among them was German physicist Werner Heisenberg (1901–1976) who, among many other contributions to quantum mechanics, formulated a mathematical treatment of the wave nature of matter that used matrices rather than wave equations. We will deal with some specifics in later sections, but it is worth noting that de Broglie’s work was a watershed for the development of quantum mechanics. De Broglie was awarded the Nobel Prize in 1929 for his vision, as were Davisson and G. P. Thomson in 1937 for their experimental verification of de Broglie’s hypothesis.

### Electron Microscopes

One consequence or use of the wave nature of matter is found in the electron microscope. As we have discussed, there is a limit to the detail observed with any probe having a wavelength. Resolution, or observable detail, is limited to about one wavelength. Since a potential of only 54 V can produce electrons with sub-nanometer wavelengths, it is easy to get electrons with much smaller wavelengths than those of visible light (hundreds of nanometers). Electron microscopes can, thus, be constructed to detect much smaller details than optical microscopes. (See .)
There are basically two types of electron microscopes. The transmission electron microscope (TEM) accelerates electrons that are emitted from a hot filament (the cathode). The beam is broadened and then passes through the sample. A magnetic lens focuses the beam image onto a fluorescent screen, a photographic plate, or (most probably) a CCD (light sensitive camera), from which it is transferred to a computer. The TEM is similar to the optical microscope, but it requires a thin sample examined in a vacuum. However it can resolve details as small as 0.1 nm (), providing magnifications of 100 million times the size of the original object. The TEM has allowed us to see individual atoms and structure of cell nuclei.
The scanning electron microscope (SEM) provides images by using secondary electrons produced by the primary beam interacting with the surface of the sample (see ). The SEM also uses magnetic lenses to focus the beam onto the sample. However, it moves the beam around electrically to “scan” the sample in the x and y directions. A CCD detector is used to process the data for each electron position, producing images like the one at the beginning of this chapter. The SEM has the advantage of not requiring a thin sample and of providing a 3-D view. However, its resolution is about ten times less than a TEM.
Electrons were the first particles with mass to be directly confirmed to have the wavelength proposed by de Broglie. Subsequently, protons, helium nuclei, neutrons, and many others have been observed to exhibit interference when they interact with objects having sizes similar to their de Broglie wavelength. The de Broglie wavelength for massless particles was well established in the 1920s for photons, and it has since been observed that all massless particles have a de Broglie wavelength  The wave nature of all particles is a universal characteristic of nature. We shall see in following sections that implications of the de Broglie wavelength include the quantization of energy in atoms and molecules, and an alteration of our basic view of nature on the microscopic scale. The next section, for example, shows that there are limits to the precision with which we may make predictions, regardless of how hard we try. There are even limits to the precision with which we may measure an object’s location or energy.

### Test Prep for AP Courses

### Section Summary

1. Particles of matter also have a wavelength, called the de Broglie wavelength, given by , where  is momentum.
2. Matter is found to have the same interference characteristics as any other wave.

### Conceptual Questions

### Problems & Exercises


# Quantum Physics

## Probability: The Heisenberg Uncertainty Principle

### Learning Objectives

By the end of this section, you will be able to:
1. Use both versions of Heisenberg’s uncertainty principle in calculations.
2. Explain the implications of Heisenberg’s uncertainty principle for measurements.

### Probability Distribution

Matter and photons are waves, implying they are spread out over some distance. What is the position of a particle, such as an electron? Is it at the center of the wave? The answer lies in how you measure the position of an electron. Experiments show that you will find the electron at some definite location, unlike a wave. But if you set up exactly the same situation and measure it again, you will find the electron in a different location, often far outside any experimental uncertainty in your measurement. Repeated measurements will display a statistical distribution of locations that appears wavelike. (See .)
After de Broglie proposed the wave nature of matter, many physicists, including Schrödinger and Heisenberg, explored the consequences. The idea quickly emerged that, because of its wave character, a particle’s trajectory and destination cannot be precisely predicted for each particle individually. However, each particle goes to a definite place (as illustrated in ). After compiling enough data, you get a distribution related to the particle’s wavelength and diffraction pattern. There is a certain probability of finding the particle at a given location, and the overall pattern is called a probability distribution. Those who developed quantum mechanics devised equations that predicted the probability distribution in various circumstances.
It is somewhat disquieting to think that you cannot predict exactly where an individual particle will go, or even follow it to its destination. Let us explore what happens if we try to follow a particle. Consider the double-slit patterns obtained for electrons and photons in . First, we note that these patterns are identical, following , the equation for double-slit constructive interference developed in Photon Energies and the Electromagnetic Spectrum, where  is the slit separation and  is the electron or photon wavelength.
Both patterns build up statistically as individual particles fall on the detector. This can be observed for photons or electrons—for now, let us concentrate on electrons. You might imagine that the electrons are interfering with one another as any waves do. To test this, you can lower the intensity until there is never more than one electron between the slits and the screen. The same interference pattern builds up! This implies that a particle’s probability distribution spans both slits, and the particles actually interfere with themselves. Does this also mean that the electron goes through both slits? An electron is a basic unit of matter that is not divisible. But it is a fair question, and so we should look to see if the electron traverses one slit or the other, or both. One possibility is to have coils around the slits that detect charges moving through them. What is observed is that an electron always goes through one slit or the other; it does not split to go through both. But there is a catch. If you determine that the electron went through one of the slits, you no longer get a double slit pattern—instead, you get single slit interference. There is no escape by using another method of determining which slit the electron went through. Knowing the particle went through one slit forces a single-slit pattern. If you do not observe which slit the electron goes through, you obtain a double-slit pattern.

### Heisenberg Uncertainty

How does knowing which slit the electron passed through change the pattern? The answer is fundamentally important—measurement affects the system being observed. Information can be lost, and in some cases it is impossible to measure two physical quantities simultaneously to exact precision. For example, you can measure the position of a moving electron by scattering light or other electrons from it. Those probes have momentum themselves, and by scattering from the electron, they change its momentum in a manner that loses information. There is a limit to absolute knowledge, even in principle.
It was Werner Heisenberg who first stated this limit to knowledge in 1929 as a result of his work on quantum mechanics and the wave characteristics of all particles. (See ). Specifically, consider simultaneously measuring the position and momentum of an electron (it could be any particle). There is an uncertainty in position  that is approximately equal to the wavelength of the particle. That is,
As discussed above, a wave is not located at one point in space. If the electron’s position is measured repeatedly, a spread in locations will be observed, implying an uncertainty in position . To detect the position of the particle, we must interact with it, such as having it collide with a detector. In the collision, the particle will lose momentum. This change in momentum could be anywhere from close to zero to the total momentum of the particle, . It is not possible to tell how much momentum will be transferred to a detector, and so there is an uncertainty in momentum , too. In fact, the uncertainty in momentum may be as large as the momentum itself, which in equation form means that
The uncertainty in position can be reduced by using a shorter-wavelength electron, since . But shortening the wavelength increases the uncertainty in momentum, since . Conversely, the uncertainty in momentum can be reduced by using a longer-wavelength electron, but this increases the uncertainty in position. Mathematically, you can express this trade-off by multiplying the uncertainties. The wavelength cancels, leaving
So if one uncertainty is reduced, the other must increase so that their product is .
With the use of advanced mathematics, Heisenberg showed that the best that can be done in a simultaneous measurement of position and momentum is
This is known as the Heisenberg uncertainty principle. It is impossible to measure position  and momentum  simultaneously with uncertainties  and  that multiply to be less than . Neither uncertainty can be zero. Neither uncertainty can become small without the other becoming large. A small wavelength allows accurate position measurement, but it increases the momentum of the probe to the point that it further disturbs the momentum of a system being measured. For example, if an electron is scattered from an atom and has a wavelength small enough to detect the position of electrons in the atom, its momentum can knock the electrons from their orbits in a manner that loses information about their original motion. It is therefore impossible to follow an electron in its orbit around an atom. If you measure the electron’s position, you will find it in a definite location, but the atom will be disrupted. Repeated measurements on identical atoms will produce interesting probability distributions for electrons around the atom, but they will not produce motion information. The probability distributions are referred to as electron clouds or orbitals. The shapes of these orbitals are often shown in general chemistry texts and are discussed in The Wave Nature of Matter Causes Quantization.
Why don’t we notice Heisenberg’s uncertainty principle in everyday life? The answer is that Planck’s constant is very small. Thus the lower limit in the uncertainty of measuring the position and momentum of large objects is negligible. We can detect sunlight reflected from Jupiter and follow the planet in its orbit around the Sun. The reflected sunlight alters the momentum of Jupiter and creates an uncertainty in its momentum, but this is totally negligible compared with Jupiter’s huge momentum. The correspondence principle tells us that the predictions of quantum mechanics become indistinguishable from classical physics for large objects, which is the case here.

### Heisenberg Uncertainty for Energy and Time

There is another form of Heisenberg’s uncertainty principle for simultaneous measurements of energy and time. In equation form,
where  is the uncertainty in energy and  is the uncertainty in time. This means that within a time interval , it is not possible to measure energy precisely—there will be an uncertainty  in the measurement. In order to measure energy more precisely (to make  smaller), we must increase . This time interval may be the amount of time we take to make the measurement, or it could be the amount of time a particular state exists, as in the next .
The uncertainty principle for energy and time can be of great significance if the lifetime of a system is very short. Then  is very small, and  is consequently very large. Some nuclei and exotic particles have extremely short lifetimes (as small as ), causing uncertainties in energy as great as many GeV (). Stored energy appears as increased rest mass, and so this means that there is significant uncertainty in the rest mass of short-lived particles. When measured repeatedly, a spread of masses or decay energies are obtained. The spread is . You might ask whether this uncertainty in energy could be avoided by not measuring the lifetime. The answer is no. Nature knows the lifetime, and so its brevity affects the energy of the particle. This is so well established experimentally that the uncertainty in decay energy is used to calculate the lifetime of short-lived states. Some nuclei and particles are so short-lived that it is difficult to measure their lifetime. But if their decay energy can be measured, its spread is , and this is used in the uncertainty principle () to calculate the lifetime .
There is another consequence of the uncertainty principle for energy and time. If energy is uncertain by , then conservation of energy can be violated by  for a time . Neither the physicist nor nature can tell that conservation of energy has been violated, if the violation is temporary and smaller than the uncertainty in energy. While this sounds innocuous enough, we shall see in later chapters that it allows the temporary creation of matter from nothing and has implications for how nature transmits forces over very small distances.
Finally, note that in the discussion of particles and waves, we have stated that individual measurements produce precise or particle-like results. A definite position is determined each time we observe an electron, for example. But repeated measurements produce a spread in values consistent with wave characteristics. The great theoretical physicist Richard Feynman (1918–1988) commented, “What there are, are particles.” When you observe enough of them, they distribute themselves as you would expect for a wave phenomenon. However, what there are as they travel we cannot tell because, when we do try to measure, we affect the traveling.

### Section Summary

1. Matter is found to have the same interference characteristics as any other wave.
2. There is now a probability distribution for the location of a particle rather than a definite position.
3. Another consequence of the wave character of all particles is the Heisenberg uncertainty principle, which limits the precision with which certain physical quantities can be known simultaneously. For position and momentum, the uncertainty principle is , where  is the uncertainty in position and  is the uncertainty in momentum.
4. For energy and time, the uncertainty principle is  where  is the uncertainty in energy and  is the uncertainty in time.
5. These small limits are fundamentally important on the quantum-mechanical scale.

### Conceptual Questions

### Problems & Exercises


# Quantum Physics

## The Particle-Wave Duality Reviewed

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the concept of particle-wave duality, and its scope.
Particle-wave duality—the fact that all particles have wave properties—is one of the cornerstones of quantum mechanics. We first came across it in the treatment of photons, those particles of EM radiation that exhibit both particle and wave properties, but not at the same time. Later it was noted that particles of matter have wave properties as well. The dual properties of particles and waves are found for all particles, whether massless like photons, or having a mass like electrons. (See .)
There are many submicroscopic particles in nature. Most have mass and are expected to act as particles, or the smallest units of matter. All these masses have wave properties, with wavelengths given by the de Broglie relationship . So, too, do combinations of these particles, such as nuclei, atoms, and molecules. As a combination of masses becomes large, particularly if it is large enough to be called macroscopic, its wave nature becomes difficult to observe. This is consistent with our common experience with matter.
Some particles in nature are massless. We have only treated the photon so far, but all massless entities travel at the speed of light, have a wavelength, and exhibit particle and wave behaviors. They have momentum given by a rearrangement of the de Broglie relationship, . In large combinations of these massless particles (such large combinations are common only for photons or EM waves), there is mostly wave behavior upon detection, and the particle nature becomes difficult to observe. This is also consistent with experience. (See .)
The particle-wave duality is a universal attribute. It is another connection between matter and energy. Not only has modern physics been able to describe nature for high speeds and small sizes, it has also discovered new connections and symmetries. There is greater unity and symmetry in nature than was known in the classical era—but they were dreamt of. A beautiful poem written by the English poet William Blake some two centuries ago contains the following four lines:
To see the World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour

### Integrated Concepts

The problem set for this section involves concepts from this chapter and several others. Physics is most interesting when applied to general situations involving more than a narrow set of physical principles. For example, photons have momentum, hence the relevance of Linear Momentum and Collisions. The following topics are involved in some or all of the problems in this section:
1. Dynamics: Newton’s Laws of Motion
2. Work, Energy, and Energy Resources
3. Linear Momentum and Collisions
4. Heat and Heat Transfer Methods
5. Electric Potential and Electric Field
6. Electric Current, Resistance, and Ohm’s Law
7. Wave Optics
8. Special Relativity
 illustrates how these strategies are applied to an integrated-concept problem.

### Test Prep for AP Courses

### Section Summary

1. The particle-wave duality refers to the fact that all particles—those with mass and those without mass—have wave characteristics.
2. This is a further connection between mass and energy.

### Conceptual Questions

### Problems & Exercises


# Atomic Physics

## Connection for AP® Courses

Have you ever wondered how we know the composition of the Sun? After all, we cannot travel there to physically collect a sample due to the extreme conditions. Fortunately, our understanding of the internal structure of atoms gives us the tools to identify the elements in the Sun’s outer layers due to an atomic “fingerprint” in the Sun’s spectrum. You will learn about atoms and their substructures, as well as how these substructures determine the behavior of the atom, such as the absorption and emission of energy by electrons within an atom.
You will learn the stories of how we discovered the various properties of an atom (Essential Knowledge 1.A.4) through clever and imaginative experimentation (such as the Millikan oil drop experiment) and interpretation (such as Brownian motion). You will also learn about the probabilistic description we use to describe the nature of electrons (Essential Knowledge 7.C.1). At this scale, electrons can be thought of as discrete particles, but they also behave in a way that is consistent with a wave model of matter (Enduring Understanding 7.C). You will learn how we use the wave model to understand the energy levels in an atom (Essential Knowledge 7.C.2) and the properties of electrons.
The content in this chapter supports:
Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure.
Enduring Understanding 1.A The internal structure of a system determines many properties of the system.
Essential Knowledge 1.A.4 Atoms have internal structures that determine their properties.
Essential Knowledge 1.A.5 Systems have properties determined by the properties and interactions of their constituent atomic and molecular substructures.
Enduring Understanding 1.B Electric charge is a property of an object or system that affects its interactions with other objects or systems containing charge.
Essential Knowledge 1.B.3 The smallest observed unit of charge that can be isolated is the electron charge, also known as the elementary charge.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.8 Energy transfer occurs when photons are absorbed or emitted, for example, by atoms or nuclei.
Big Idea 7 The mathematics of probability can be used to describe the behavior of complex systems and to interpret the behavior of quantum mechanical systems.
Enduring Understanding 7.C At the quantum scale, matter is described by a wave function, which leads to a probabilistic description of the microscopic world.
Essential Knowledge 7.C.1 The probabilistic description of matter is modeled by a wave function, which can be assigned to an object and used to describe its motion and interactions. The absolute value of the wave function is related to the probability of ﬁnding a particle in some spatial region.
Essential Knowledge 7.C.2 The allowed states for an electron in an atom can be calculated from the wave model of an electron.
Essential Knowledge 7.C.4 Photon emission and absorption processes are described by probability.


# Atomic Physics

## Discovery of the Atom

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the basic structure of the atom, the substructure of all matter.
How do we know that atoms are really there if we cannot see them with our eyes? A brief account of the progression from the proposal of atoms by the Greeks to the first direct evidence of their existence follows.
People have long speculated about the structure of matter and the existence of atoms. The earliest significant ideas to survive are due to the ancient Greeks in the fifth century BCE, especially those of the philosophers Leucippus and Democritus. (There is some evidence that philosophers in both India and China made similar speculations, at about the same time.) They considered the question of whether a substance can be divided without limit into ever smaller pieces. There are only a few possible answers to this question. One is that infinitesimally small subdivision is possible. Another is what Democritus in particular believed—that there is a smallest unit that cannot be further subdivided. Democritus called this the atom. We now know that atoms themselves can be subdivided, but their identity is destroyed in the process, so the Greeks were correct in a respect. The Greeks also felt that atoms were in constant motion, another correct notion.
The Greeks and others speculated about the properties of atoms, proposing that only a few types existed and that all matter was formed as various combinations of these types. The famous proposal that the basic elements were earth, air, fire, and water was brilliant, but incorrect. The Greeks had identified the most common examples of the four states of matter (solid, gas, plasma, and liquid), rather than the basic elements. More than 2000 years passed before observations could be made with equipment capable of revealing the true nature of atoms.
Over the centuries, discoveries were made regarding the properties of substances and their chemical reactions. Certain systematic features were recognized, but similarities between common and rare elements resulted in efforts to transmute them (lead into gold, in particular) for financial gain. Secrecy was endemic. Alchemists discovered and rediscovered many facts but did not make them broadly available. As the Middle Ages ended, alchemy gradually faded, and the science of chemistry arose. It was no longer possible, nor considered desirable, to keep discoveries secret. Collective knowledge grew, and by the beginning of the 19th century, an important fact was well established—the masses of reactants in specific chemical reactions always have a particular mass ratio. This is very strong indirect evidence that there are basic units (atoms and molecules) that have these same mass ratios. The English chemist John Dalton (1766–1844) did much of this work, with significant contributions by the Italian physicist Amedeo Avogadro (1776–1856). It was Avogadro who developed the idea of a fixed number of atoms and molecules in a mole, and this special number is called Avogadro’s number in his honor. The Austrian physicist Johann Josef Loschmidt was the first to measure the value of the constant in 1865 using the kinetic theory of gases.
Knowledge of the properties of elements and compounds grew, culminating in the mid-19th-century development of the periodic table of the elements by Dmitri Mendeleev (1834–1907), the great Russian chemist. Mendeleev proposed an ingenious array that highlighted the periodic nature of the properties of elements. Believing in the systematics of the periodic table, he also predicted the existence of then-unknown elements to complete it. Once these elements were discovered and determined to have properties predicted by Mendeleev, his periodic table became universally accepted.
Also during the 19th century, the kinetic theory of gases was developed. Kinetic theory is based on the existence of atoms and molecules in random thermal motion and provides a microscopic explanation of the gas laws, heat transfer, and thermodynamics (see Introduction to Temperature, Kinetic Theory, and the Gas Laws and Introduction to Laws of Thermodynamics). Kinetic theory works so well that it is another strong indication of the existence of atoms. But it is still indirect evidence—individual atoms and molecules had not been observed. There were heated debates about the validity of kinetic theory until direct evidence of atoms was obtained.
The first truly direct evidence of atoms is credited to Robert Brown, a Scottish botanist. In 1827, he noticed that tiny pollen grains suspended in still water moved about in complex paths. This can be observed with a microscope for any small particles in a fluid. The motion is caused by the random thermal motions of fluid molecules colliding with particles in the fluid, and it is now called Brownian motion. (See .) Statistical fluctuations in the numbers of molecules striking the sides of a visible particle cause it to move first this way, then that. Although the molecules cannot be directly observed, their effects on the particle can be. By examining Brownian motion, the size of molecules can be calculated. The smaller and more numerous they are, the smaller the fluctuations in the numbers striking different sides.
It was Albert Einstein who, starting in his epochal year of 1905, published several papers that explained precisely how Brownian motion could be used to measure the size of atoms and molecules. (In 1905 Einstein created special relativity, proposed photons as quanta of EM radiation, and produced a theory of Brownian motion that allowed the size of atoms to be determined. All of this was done in his spare time, since he worked days as a patent examiner. Any one of these very basic works could have been the crowning achievement of an entire career—yet Einstein did even more in later years.) Their sizes were only approximately known to be

  , based on a comparison of latent heat of vaporization and surface tension made in about 1805 by Thomas Young of double-slit fame and the famous astronomer and mathematician Simon Laplace.
Using Einstein’s ideas, the French physicist Jean-Baptiste Perrin (1870–1942) carefully observed Brownian motion; not only did he confirm Einstein’s theory, he also produced accurate sizes for atoms and molecules. Since molecular weights and densities of materials were well established, knowing atomic and molecular sizes allowed a precise value for Avogadro’s number to be obtained. (If we know how big an atom is, we know how many fit into a certain volume.) Perrin also used these ideas to explain atomic and molecular agitation effects in sedimentation, and he received the 1926 Nobel Prize for his achievements. Most scientists were already convinced of the existence of atoms, but the accurate observation and analysis of Brownian motion was conclusive—it was the first truly direct evidence.
A huge array of direct and indirect evidence for the existence of atoms now exists. For example, it has become possible to accelerate ions (much as electrons are accelerated in cathode-ray tubes) and to detect them individually as well as measure their masses (see More Applications of Magnetism for a discussion of mass spectrometers). Other devices that observe individual atoms, such as the scanning tunneling electron microscope, will be discussed elsewhere. (See .) All of our understanding of the properties of matter is based on and consistent with the atom. The atom’s substructures, such as electron shells and the nucleus, are both interesting and important. The nucleus in turn has a substructure, as do the particles of which it is composed. These topics, and the question of whether there is a smallest basic structure to matter, will be explored in later parts of the text.

### Section Summary

1. Atoms are the smallest unit of elements; atoms combine to form molecules, the smallest unit of compounds.
2. The first direct observation of atoms was in Brownian motion.
3. Analysis of Brownian motion gave accurate sizes for atoms (

 on average) and a precise value for Avogadro’s number.

### Conceptual Questions

### Problems & Exercises