Special Relativity and Quantum Physics

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and consistently found to be correct, special relativity forms a framework on which ... view of nature demanded by quantum physics, illustrated by a revisiting of ...... In practice, every such measurement will have an associated uncertainty ... there is no conflict of this example with the uncertainty principle, we need to exam-.
Special Relativity and Quantum Physics

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This and the next chapter bring together a number of fundamental concepts about matter and radiation, some of which we have anticipated in previous discussions when needed. The title of this chapter names the two major theories developed in the 20th century that have most revolutionized physics. Relativity and quantum physics are together sometimes known as modern physics. We begin this chapter with a brief discussion of some aspects of special relativity, a theory developed by Albert Einstein that has brought about major changes in our understanding of the world. Thoroughly tested and consistently found to be correct, special relativity forms a framework on which modern physics rests. The chapter then continues with an overview of the probabilistic view of nature demanded by quantum physics, illustrated by a revisiting of the doubleslit experiment. Some of the main features of quantum physics are then discussed, including the Schrödinger equation and the uncertainty principle. The chapter concludes with a discussion of the quantum basis of scanning tunneling microscopy, capable of viewing individual atoms. Our discussion continues in the next chapter with the quantum physics of atoms and molecules and their study by spectroscopy, including the laser which is one of the most important tools in science and medicine today.

1. SPECIAL RELATIVITY: MASS–ENERGY AND DYNAMICS Special relativity is concerned with our fundamental notions of time, space, mass, energy, and motion at constant velocities. Albert Einstein published the theory of special relativity in 1905 when he was 26 years old. In that same year he also published fundamental papers on Brownian motion and on the photoelectric effect, discussed below, for which he received the Nobel Prize. Twelve years later, in 1917, he published the theory of general relativity, which quantitatively shows the equivalence between accelerated motions and gravity, known as the equivalence principle, replacing the gravitational force with a curvature of space and time. Although Einstein’s theory of general relativity has been successfully tested and accepted today, those tests are relatively few in number and its impact on physics is much more limited than that of special relativity. One important everyday application of general relativity is a correction needed for the extremely accurate time keeping required for GPS (global positioning system; Figure 24.1); without general relativity corrections, GPS navigational errors would be about 10 km per day. Special relativity, on the other hand, has been thoroughly tested and is completely ingrained in all areas of modern physics. Relativity is often thought to be mathematically complex, but it is only general relativity, not discussed here, that involves higher mathematics. Special relativity can be explained without the use of much mathematics and so can be understood by the nonscientist, but it involves ideas that seem contrary to our intuition. We live in a world of extremely slow moving objects compared to the speed of light. Relativity (from now on we omit the word “special” because we limit our discussion to special relativity) deals J. Newman, Physics of the Life Sciences, DOI: 10.1007/978-0-387-77259-2_24, © Springer Science+Business Media, LLC 2008

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FIGURE 24.1 A handheld GPS only works with corrections from general relativity.

with new phenomena that occur at speeds approaching the speed of light. We have no intuitive basis for understanding such processes since we never experience motion at such speeds. Even a plane traveling at 600 mph travels only at about 1 millionth the speed of light. All of the equations of relativity reduce to equations we have already studied when the speeds of objects are small compared to the speed of light, as we shall see. Two fundamental postulates form the basis of relativity theory from which all its consequences follow. The first, known as the principle of relativity, is that all the laws of physics are the same in all inertial frames of reference. We have already seen an example of this principle in mechanics in the form of Newton’s first law. Relative velocities may be different in two different inertial frames, however, accelerations of objects and the description of the forces acting to produce motion will be the same in all inertial reference frames. Einstein’s relativity principle extends this notion to cover all the laws of physics, not just those of mechanics. The second postulate concerns the constancy of the speed of light and states that the speed of light in vacuum has the same value c in all inertial reference frames. It is remarkable that these two postulates alone lead to the development of such a powerful theory. We limit our discussion here to those salient features of dynamics that we need later in this book, omitting the fascinating consequences of relativity on our notion of space and time. Consider a point particle of mass m, moving with a velocity v in the x-direction as seen by an observer. Classically, the momentum of the particle would be defined as p  mv  m(x/t), where x is the displacement of the particle in a time interval t. In place of this, the relativistic momentum is defined as p

mv

11  v 2 /c2

 gmv,

(24.1)

where the Lorentz factor  is defined by g

1

11  v2 /c2

.

Although for a stationary particle   1, even when the particle moves at 0.1 c, quite a large velocity, the value for  is only 1.005. Figure 24.2 shows how  varies with the ratio v/c, confined to lie between 0 and 1; note that  grows very rapidly as v approaches c. This formula can be directly generalized to three-dimensional motion by treating p and v as vectors. Note that for small values of v we can neglect the term v2/c2 in the denominator of Equation (24.1) so that the expression for momentum reduces to its classical value. As v approaches c, however, the momentum of the particle, being proportional to , increases at a much faster rate than the classical linear dependence on v.

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FIGURE 24.2 The factor  versus v/c showing its rapid rise as v approaches c.

25

20

γ

15

10

5

0 0.0

0.2

0.4

0.6

0.8

1.0

v /c

Because the momentum increases so rapidly as the particle’s velocity approaches c, it requires an ever-increasing force, equal to the rate of change of momentum, to accelerate the particle. If the particle starts from rest, an increase in its velocity by 10% of the speed of light will produce a proportional momentum increase of just about 10%. However, if the particle is already moving at half the speed of light, the change in momentum for a 0.1 c increase in velocity (a 20% increase, from 0.5 c to 0.6 c) will be about 30%, whereas if the particle is already moving at 85% of the speed of light, the corresponding increase in momentum for the same 0.1 c increase (about a 12% increase from 0.85 c to 0.95 c) will be almost 90%. As the velocity of the particle approaches c, its momentum increases very rapidly, and therefore the change in momentum needed to produce the same step increase in its velocity will also dramatically increase. Because an ever-increasing force is needed to increase the particle’s momentum, this effect prevents a material particle (one with a nonzero mass) from ever attaining a velocity equal to the speed of light. Another important variable of dynamics is the kinetic energy, classically given as KE  12 mv2. The relativistic kinetic energy expression looks quite different and is given by KE 

mc2

11  v2 /c2

 mc2  gmc2  mc2.

(24.2)

This is indeed an energy that depends on motion because if v  0, then   1 and the expression clearly reduces to KE  0. Although it is not apparent that for small velocities compared to c this reduces to the classical expression, we can show this by expanding the square root term in Equation (24.2) using the binomial theorem 

1 2

(1  x2 )  1 

x2  ..., 2

(24.3)

valid for x 1).

(24.9)

This same expression holds for ultrarelativistic massive particles, whose speeds approach c so that  >> 1, because the first term on the right in Equation (24.8) dominates and we can neglect the second rest energy term. The ideas we have developed in this section are used in the remainder of this book in various discussions of modern physics. Relativity also deals with other concepts related to motion at large constant velocities, including fundamental changes in our notion of distance and time. These we leave for the interested reader to find in any one of a large number of popular books that discuss special relativity, including one by Albert Einstein himself.

2. OVERVIEW OF QUANTUM THEORY We now take a veritable quantum leap and begin considering our current understanding of the atomic world of nature. Earlier in this book we have seen the notion of wave-particle duality, that in nature the elementary constituents of matter and radiation can appear to behave as either particles or waves, depending upon the interactions with their environment. For example, photons, the elementary quanta of radiation, can behave as waves (in interference and diffraction), or, in other situations as we soon show, photons can behave as particles. The wave packet picture was introduced in Chapter 19 as a way to visualize this duality, with the wave packet capable of collapsing to be more particlelike or expanding to be more wavelike in space depending on its interactions. Here we discuss this in more general terms and show FIGURE 24.3 The photoelectric that the picture also applies to all other elementary “particles” and sometimes even to effect. Light incident on the macroscopic systems. We discuss a series of different experiments that illustrate the photocathode electrode in a wave–particle duality nature of photons and other elementary “particles” such as vacuum tube causes electrons to be ejected and attracted to the electrons. The photoelectric effect is a very important process in which light causes the anode (by a positive potential) to make up a current measured by the emission of electrons from a metal surface. This phenomenon is the basis for a external ammeter. variety of light-detecting devices that produce electric currents in response to light. Many of the features of the interaction of light with a metal surface could not be explained on the basis of a wave theory of light and these led photocathode anode Albert Einstein to propose a theory of the photoelectric effect in 1905 based on photons. When light is directed on a metal cathode (negative electrode) within a vacuum tube, as shown in Figure 24.3, an electric current can be generated at the anode (positive electrode) when a potential difference is applied across the electrodes to collect the emitted electrons, even though there is no wire connected between the two electrodes. According to the wave theory of light, A the intensity of light should be proportional to the beam energy, and for a

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sufficiently intense beam, no matter what the wavelength, one should expect electrons to be ejected from the metal surface after gaining energy from the light. Indeed, for shorter wavelength light, the electric current is proportional to the intensity of the light. However, if the wavelength of the light is long enough, then regardless of the intensity of the beam or the applied voltage supplied by the battery no electrons are generated. Classical wave physics is unable to explain the conditions when such an electric current will appear or will not appear. Einstein’s explanation of the photoelectric effect is based on light consisting of individual photons, each with an energy given by (see Chapter 19) E  hf 

hc , l

(24.10)

where h is Planck’s constant, h  6.63  1034 J-s, and we have used the fact that c  f . Photons also carry a momentum, according to Equation (24.9), given by p

E h  . c l

(24.11)

Equations (24.10) and (24.11) relate the photon energy and momentum, particlelike properties, to the wavelike properties of wavelength or frequency. If the wavelength of the light is longer than some threshold value, then the energy of each photon will be too low to provide the minimum energy necessary to eject an electron from the metal surface, an energy known as the work function . In this case, no electrons will be ejected.1 When the photon energy exceeds the work function, a single photon can interact with an atom in the metal surface and eject a single electron. Those electrons that do escape from the “photocathode” surface can be attracted to the anode, by applying a positive potential difference between the electrodes, and make up the detected current. The amount of current is then proportional to the number of photons per second in the beam, this being proportional to the intensity of the beam. Beam intensity is defined as the energy per unit time per cross-sectional area and for a monochromatic beam is determined by the product of the energy of each photon and the number of such photons per second per cross-sectional area. Now, depending on the wavelength of the incident light, emitted electrons will have more or less kinetic energy. In order to measure the kinetic energy of the emitted electrons, the polarity of the applied voltage can be reversed so that the electrons will be repelled by the anode. When the most energetic electrons are just stopped by this reversed voltage, known as the stopping potential, we know that KEmax  eVstop,

(24.12)

and such a measurement can determine the maximum kinetic energy of the electrons emitted in the photoelectric effect. Einstein predicted that this maximum kinetic energy would be given by KEmax  hf  £,

(24.13)

so that the excess photon energy above the minimum energy needed to escape from the surface, the work function, equals the maximum kinetic energy. Electrons requiring more energy to escape from the surface will be left with less kinetic energy. Because kinetic energy must be positive, this relation implies that there is a minimum 1Strictly speaking, we now know this to be untrue: if the light source is a high-power laser, then there can be such an enormous number of photons that there is a nonnegligible probability that a single electron can absorb two or more subthreshold energy photons simultaneously and gain sufficient energy to escape. This is similar to the basis of multiphoton microscopy discussed at the end of Section 1 of the previous chapter.

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frequency of light that is needed for electrons to just escape from the metal surface with essentially no kinetic energy given by fmin 

£ . h

(24.14)

Equation (24.13) also correctly predicts that the maximum kinetic energy of the electrons depends only on the frequency and is independent of the intensity of the light.

Example 24.2 Suppose that red light of   633 nm or blue light of 488 nm is directed on a photocathode with a work function of 2.25 eV. If 1012 photons per second of each color are separately incident on the photocathode, what will be the detected photocurrent in each case assuming 100% efficiency and all the emitted photoelectrons are captured by the anode? If the intensity of each light beam is increased by a factor of 10, what will happen? What is the stopping potential in each case? Solution: The energy of the red and blue photons are given by hc/ and are equal to (after converting to eV) 2.0 and 2.5 eV, respectively. Therefore, given the work function of 2.25 eV, red photons have insufficient energy to eject electrons whereas blue photons will each lead to an electron being detected at the anode (given the assumed 100% efficiencies) leading to a photocurrent corresponding to 1012 electrons per second or a current of (1012 e/s) (1.6  1019 C/e)  1.6  107 A  0.16 A. If the intensities are increased by a factor of 10 there will still be no emitted electrons with the red beam because the individual photon energy has not changed, and the photocurrent detected using the blue beam will increase by a factor of 10 to 1.6 A. The stopping potential for the red beam experiment is zero because no electrons are detected at all whereas for the blue beam experiment, because the electrons are emitted with a maximum kinetic energy of 2.5  2.25  0.25 eV, the stopping potential will be 0.25 V. Note carefully the units here.

A second experiment that demonstrates the particlelike nature of photons is the scattering of x-rays, high-energy photons, by the electrons of a material. In the early 1920s Arthur Compton discovered that the wavelength of x-rays gets slightly longer after scattering from a graphite target. He discovered that the process, now known as Compton scattering, could be completely explained by assuming that the x-rays carried energy and momentum given by Equations (24.10) and (24.11) and that the scattering simply conserved kinetic energy and momentum. Such an elastic collision is analyzed in a straightforward way using energy and momentum conservation in two dimensions just as it would be for billiard balls on a frictionless table. The resulting shift to longer x-ray wavelengths is due to the electron, initially at rest, gaining some momentum and kinetic energy at the expense of the photon (see Figure 24.4).

λ + Δλ

λ

x-ray

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θ

scattered x-ray

FIGURE 24.4 Compton scattering of an x-ray photon by an electron. The scattered photon with longer wavelength and the recoil electron are shown dotted.

scattered electron

587

A decreased photon momentum or energy has an associated increase in wavelength, known as the Compton wavelength shift, . Using energy and momentum conservation, Compton derived a formula for the wavelength shift ¢l  lc (1 cos u),

(24.15)

where is the scattering angle and c is the Compton wavelength of the electron, a fundamental constant given by lc 

h  2.43  10 12 m, mc

where m is the mass of an electron. Thus, the Compton shift vanishes for forward scattering, where the scattering angle is close to 0° indicating little interaction between the x-ray and electron, and is a maximum for backscattering when equals 180° and the x-ray has strongly interacted with the electron. We mention that both Compton scattering and the photoelectric effect are important in the making of a medical x-ray, the first in the x-ray/body interaction and the second in the detection process. Having just studied two of the important experiments establishing the particlelike nature of photons under certain conditions, let’s reconsider the double-slit interference experiment for light discussed earlier in Chapter 22 where we treated light as a wave. Imagine that we reduce the intensity of the light source so low that only one photon at a time arrives at the slits. Figure 24.5 shows the experiment. It is found that individual photons are detected at the screen at localized spots implying that the photon wave packet “collapses” when detected. However, after many such detections, the pattern of the total detected intensity is the same as that observed directly at higher light levels. In other words, even though individual detection events are localized on the screen, no photons ever arrive at positions on the screen that correspond to destructive interference bands whereas many more photons than the average arrive at the positions of constructive interference, according to the path difference equations of Chapter 22. If each individual photon went through one slit or the other, we would not expect to see an interference pattern because, with only one photon at a time, there would be no interference occurring. We must conclude that the individual photons are going through both slits and interfering with themselves, with their own wave packet. Given our (brief) discussion of wave packets and the notion of diffraction, it is not impossible to accept this notion. Individual wave packets, representing each photon, must travel through both slits, diffract at each, and recombine according to the rules of interference. When subsequently detected at the detector in the far-field, the wave packets must collapse and interact with the atoms of the detector as a “particle” getting detected at one particular location. Amazingly, if the same experiment were to be done with electrons (but using different detection equipment), we would observe a similar result. The pattern of detected electrons on a screen far from the double-slits would be that produced by an interference pattern of waves using a wavelength for the electron given by the same expression as

FIGURE 24.5 The double-slit experiment at very low light levels so that individual photons are detected. A long experiment detecting many photons will build up a multiple exposure that is identical to that detected at higher light levels. The necessary conclusion is that individual photons interfere with themselves in passing through both slits.

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Equation (24.11),   h/p, known as the de Broglie wavelength of the electron. The electrons could be detected, for example, by having them strike a fluorescent screen emitting localized flashes of light. The slit separation would need to be made comparable to the de Broglie wavelength of the electron, but by adjustment of the electron’s momentum this even can be matched to the same slit size as used for the photon experiment. At high electron beam intensities, an interference pattern would be directly observed on the screen. At very low electron beam intensity, with individual electrons arriving at the double-slit, the same interference pattern would be observed after an extended time exposure, again forcing us to conclude that each electron went through both slits simultaneously and interfered with itself. This seems at first sight to be inconceivable because the electron is known to be a fundamental “particle” that has no internal structure and is not divisible into subpieces. Despite our difficulties in accepting this, the electron does indeed behave as a wave, known as a matter wave. Although proposed much earlier and often used as a conceptual argument, this double-slit experiment with individual electrons was actually performed first in 1961 and has been verified in many ways since. The first experiment to verify the wave nature of the electron was done by Davisson and Germer in 1927. By studying the diffraction of a beam of electrons from a crystal and observing ring patterns of maxima and minima, these experiments were able to verify the correctness of the de Broglie relation for the wavelength of the electron. Electrons, as well as photons, are said to exhibit wave–particle duality, sometimes behaving as a wave, as in situations showing diffraction and interference effects, and sometimes behaving as a particle, as in the detection process where particle mechanics concepts of momentum and energy “packets” apply. Our conclusions for electrons also hold for all other elementary particles, each having its own de Broglie wavelength, depending on its momentum. Such wavelike effects of matter are not normally observed for macroscopic matter because the de Broglie wavelengths become extremely tiny. For example, a 1 kg mass traveling at 1 m/s has a de Broglie wavelength of about 1033 m, much too small to produce any observable wave effects. But in the world of elementary particles, the masses are tiny, so that de Broglie wavelengths are large enough to produce dramatic effects. Even nonrelativistic electrons, accelerated through a potential difference of 1 V, have a momentum of p  12mE  5.4  10 25 kg m/s, and a corresponding de Broglie wavelength of 1.2 nm. This wavelength is large compared to atomic dimensions and such slow moving electrons can therefore be expected to exhibit diffraction and interference effects when interacting with a crystalline array of atoms, just as light does with an array of slits. Figure 24.6 shows an example of an electron diffraction pattern. In addition to mass and electric charge, each electron carries another intrinsic property called spin. Just as mass creates gravity and charge creates the electric force, FIGURE 24.6 Electron diffraction pattern from a thin germanium crystal.

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spin creates an interaction as well, another kind of repulsive force between electrons. Unlike gravity and the electric force, though, we can’t write down a specific equation for this interaction. Instead, it is expressed as a rule: no two electrons occupying the same region of space can be in exactly the same state of motion (or have the same set of quantum numbers). This rule is called the Pauli exclusion principle, and, among other things, it is responsible for the great variety of chemical differences we observe among atoms. We study this further in the next chapter where we show in more detail that this is responsible for the different known types of atoms. Here we need to point out that this principle only applies to particles with half-integral spin. Some macroscopic systems also exhibit quantum mechanical effects; particularly notable examples are superconductors and superfluids. In some materials at sufficiently low temperature, the conduction electrons pair up so that these “Cooper pairs” have integral spin and are no longer subject to the Pauli exclusion principle. They are all able to occupy the same low energy state and not interact with the material lattice around them. In this case their electrical resistance is, in fact, equal to zero. These materials are called superconductors and a variety of different types of materials have been discovered that become superconductors at sufficiently low temperatures. Superconducting wires are used in large electromagnets to produce very large magnetic fields without heating problems when their temperature is sufficiently low, typically at liquid helium temperatures of about 4 K. For example, these superconducting magnets are used in MRI facilities in hospitals. Such superconductors eliminate I2R heating and once a current is established in these materials, it persists without the need for a continual energy supply such as a battery or power supply. A major goal of this area of research is to develop materials that are superconducting at ambient, or near ambient, temperatures and that can be fabricated into wires or other types of conductors to avoid the costs of maintaining those extremely low temperatures. An analogous situation can occur in certain fluids when they are cooled to very low temperatures. For example, when 4He, with paired protons, neutrons, and electrons, is cooled below 2.18 K, it becomes a superfluid with very unusual properties. Superfluids have no viscosity, so that a particle traveling through them moves with no friction. Such superfluids can also flow through microscopic pores and channels that would not be accessible to normal fluids because of surface tension. 3He can also behave as a superfluid at about 1000 times colder temperatures, in a mechanism similar to superconductors, by forming “Cooper pairs” of 3He which behave as integral spin particles, so that they are not subject to the Pauli exclusion principle. Superfluidity is very rare and has only been found in a handful of systems other than helium.

3. WAVE FUNCTIONS; THE SCHRÖDINGER EQUATION We’ve seen that photons and other elementary particles such as the electron have both wavelike and particlelike properties that are related to each other. For example, treating light as made of photons, particles of zero rest mass, its energy E and momentum p are connected through the relation E  pc. But these quantities are connected with the wavelike properties of frequency and wavelength through Equations (24.10) and (24.11). Furthermore, viewing light as an electromagnetic wave, we’ve seen that the intensity, or energy per unit area per unit time, is proportional to the square of the electric field. How are these two pictures related to each other? In our rediscovery of the double-slit experiment for single photons we just saw that the photon wave packet is a representation of the spatial extent of the photon. This implies that the square of the electric field must be a measure of where the photon is located (see below). Knowing that electrons and other elementary particles also exhibit both particlelike and wavelike behavior, scientists were prompted to look for a wave theory of matter. But in that case what is it that is waving; whose square is related to the electron’s whereabouts? Quantum mechanics, developed in the 1920s, introduces a wave function that is dependent on both time and position, and that represents all the possible information

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obtainable about an elementary particle or system of particles under study. Note our mix of the words particle and wave function in the same description of the system. An electron, for example, is described completely by its wave function.

The square of the wave function for the electron, 2(x, y, z, t), multiplied by the volume of a small region in space V located at (x, y, z), represents the probability that the electron will be found within that volume at that position at the specified time ° 2 (x, y, z, t)¢V  Probability to find electron within ¢V at (x, y, z) at time t.

(24.16)

According to this definition 2 represents a probability density, or probability per unit volume. Depending on the dimensionality of a particular problem, we might replace the volume with the surface area or simply the linear distance. For example, in our description of the double-slit experiment with an electron, with x the distance along the screen measured from the central axis, 2(x, t) x would represent the probability of finding an electron within a distance x at position x at time t. This probability will have the same spatial variation as the interference patterns with light discussed in the last chapter. Locations of complete destructive interference would have 2  0, and interference maxima would correspond to maxima in 2. We can make a close analogy between for matter waves and the electric field E for photons. We know that for photons, the intensity I, proportional to E2 and representing the photon energy per unit area (or photon flux) per unit time, is also proportional to the number of photons N. If the intensity and therefore number of photons is very small, as in the low-intensity double-slit experiment discussed in the last section, then we can interpret E2x, evaluated at some point on the screen, as the probability that a photon will be detected within x at that point on the screen. Similarly for an electron, for example, 2V, evaluated at a point represents the probability of finding an electron within the small volume V at that point. Because we can interpret 2 V as the probability of finding the electron within V, and it is also clear that the electron must be found somewhere within the confines of the system boundary (with certainty, or with a probability of 1), we must have that g ° 2 ¢V  1,

(24.17)

where the summation is over all the volume available in the system. This is known as the normalization condition and establishes the scale for quantifying . Quantum mechanics provides an equation, the Schrödinger equation, which plays the same role as Maxwell’s equations play in electromagnetism (see the box below). Schrödinger’s equation allows one to compute the space- and time-dependence of the wave function for any quantum system. Only wave functions for simple systems can be analytically determined; those for complicated systems of many bodies must be approximated and calculated using computers. To give a sense of the nature of wave functions, let’s consider the problem of a particle trapped in a box. We consider a one-dimensional problem, with a particle bouncing back and forth between end walls, only experiencing a force at the walls where we imagine the potential energy to rise infinitely steeply as shown in Figure 24.7. The E

Ψ2

E3 = 9E1 E2 = 4E1 E1 x

FIGURE 24.7 A quantum mechanical particle in a one-dimensional box. The first three wave functions are shown, each having a discrete energy shown on the right (color coded); the longer wavelengths correspond to lower energy states as discussed in the text.

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fundamental concept invoked here is that the matter wave must be a standing wave within the box. Only a standing wave results in nonzero amplitudes and we show that a standing wave also leads to a discrete set of possible energy levels for the particle. A matter wave with energy different from one of those discrete energy levels would, through interference, completely cancel itself on multiple reflection within the box. This is precisely the same idea as was discussed in connection with standing waves on a string or in an air column back in Chapters 10 and 11. We also discuss this further in the next section in connection with the uncertainty principle. The standing wave expressions for (x, t) in our one-dimensional box of length L are found by applying the boundary conditions that for all time there are nodes at the ends, (0, t)  (L, t)  0. We find that the possible standing wave functions are °n (x, t)  An sin (npx/L),

(24.18)

independent of time, where An are the amplitudes of the nth harmonic of the wave (see Equation (10.18)) and are chosen according to the normalization requirement of Equation (24.17). Equation (24.18) satisfies the boundary conditions (please check this!) and gives a set of standing waves with wavelengths corresponding to n  2L/n, as can be seen by rewriting the argument of the sine function as, a A simplified form of Schrödinger’s equation, valid for a particle of mass m and energy E moving along the x-axis in a potential energy PE(x), is a time-independent differential equation for the wave function (x): -h2 8p2 m

a

d 2 °(x) dx2

b  PE(x)°(x)  E°(x).

The potential energy function represents the total of all the interactions that the particle experiences, with typical model forms for PE being square wells, barriers, Coulomb potentials, harmonic oscillator potentials, or more realistic functions representing molecular potentials (see Figure 24.8). In a straightforward fashion this equation can be generalized to three-dimensional space and applied directly, for example, to solve for the wave function of the electron in the hydrogen atom using the Coulomb potential due to the proton. The wave functions obtained give the probability density for the electron and give the mean radius of the ground state of the hydrogen atom. In solving the Schrödinger equation, with each wave function there is a corresponding energy of the electron, so that the values of E are discrete and form an energy-level diagram that we have alluded to several times in this text.

592

2px 2px b . (2L /n) ln

Using the relation between the de Broglie wavelength and the momentum p  h/, we see that the momentum of the particle is quantized and must satisfy pn  nh /2L, so that the energy of the (nonrelativistic) particle must also be quantized (En  pn2/2m) and given by En  n2

h2 . 8mL2

(24.19)

Figure 24.7 shows the first few wave functions and the corresponding energy level diagram for the particle in a one-dimensional box. The n  1 state is the ground state for this system and has an energy given by E1 

h2 . 8mL2

(24.20)

It is noteworthy that the particle cannot have zero kinetic energy according to our results, but must have at least a minimum energy given by Equation (24.20), known as the zero-point energy because the particle will have this same energy even at a temperature of absolute zero. The particle in a box problem, although perhaps not very realistic, does illustrate some of the basic ideas of quantum mechanics. Other types of potentials, shown in Figure 24.8, can be analyzed in a similar manner to give results applicable to more realistic problems. For example, the “finite square well” problem (curve a in the figure) or better the “Coulomb potential barrier” (curve b) can be used to model particles within the nucleus as we show in the next chapter. In this case, because the wall is not infinitely high, we show that although it is impossible for particles with a small energy to escape from the “well” classically, quantum mechanics predicts some possibility to penetrate the wall and escape. This phenomenon can be used to model radioactive decay of nuclei. Similarly, a particle that meets a “finite barrier potential” (curve c in the figure) with an energy smaller than the barrier should be totally reflected classically, but quantum mechanics predicts

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FIGURE 24.8 Various potential functions commonly used to represent physical situations: (a) finite square well; (b) Coulomb well; (c) finite barrier; (d) molecular potential (Lennard–Jones type); (e) simple harmonic oscillator.

PE(x)

x

a

b

c

d

e

that there will be some probability that the particle can “tunnel” through the barrier and reach the other side. This phenomenon is important in our discussion of the scanning tunneling microscope in the next section. Finally, various potential energy curves (e.g., curve d) can be used to model the interactions of valence electrons in atoms or molecules, as discussed in the next chapter.

4. UNCERTAINTY PRINCIPLE; SCANNING TUNNELING MICROSCOPE We have seen that quantum mechanical particles exhibit wave–particle duality, appearing sometimes to have exclusively wavelike and sometimes exclusively particlelike properties. Niels Bohr referred to this as the principle of complementarity. Quantum mechanics takes the view that in order to have definite knowledge of a certain parameter describing a particle, such as its position, momentum, or energy, a measurement must be performed. In practice, every such measurement will have an associated uncertainty due to, at the very least, the precision of the measuring instruments and the skill of the measurer. For example, a measurement of a particle’s position or velocity may be limited by the precision of the meter stick or of the clock used. No matter how sophisticated the measurement, there will always be limitations on the precision of the measurement. In the world of elementary quantum mechanical particles there are fundamental intrinsic limitations on the accuracy of measurements due to the interaction of the measuring instrument with the particle. Unlike the usual experimental limits on precision of a measurement, these more fundamental limitations do not depend on the precision of measurement instruments or on the skill of the measurer. If we try to determine both the position and momentum of, say, an electron, then no matter how “gentle” a measurement we make, there is always an uncertainty in precisely how the interaction occurs that is intrinsic in nature. For example, suppose we try to “see” the position of an electron by scattering a photon from it. We know that the photon has a wavelength that will fundamentally limit the resolution with which we can “see” due to diffraction effects. In the scattering process, the photon will also impart some of its energy to the electron. To better locate the position of the electron we might decrease the wavelength of the photon so that during the scattering event we may “see” with greater resolution. In so improving the precision of the electron position measurement, however, the photon’s energy and momentum increase and the electron will receive an uncertain fraction of the photon’s larger energy leading to a greater uncertainty in the electron’s momentum. This is a fundamental problem, not one that can be eliminated by more careful measurement apparatus or skill. Let’s sketch a semiquantitative analysis of the scattering event. The resolution uncertainty is comparable to the wavelength of the photon, so that ¢x L l.

(24.21)

Because the photon’s momentum is given by p  h/, and some indeterminate fraction is imparted to the electron, we also have that ¢p L

h . l

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(24.22)

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The product of uncertainty in position, x, and the uncertainty in momentum along the x-direction, p, leads to the Heisenberg uncertainty principle ¢x ¢p L h,

(24.23)

where it is understood that this expression gives the minimum uncertainty product possible.

The uncertainty principle (see Figure 24.9) tells us that if we know the exact position of a particle, so that x is zero, then we can have no knowledge at all of the particle’s momentum (p ~ ). According to our experience this principle makes no sense at first sight. We can measure the position of, say, a marble with very high precision while it sits quite at rest on a table, so that p  0 very precisely. To see why there is no conflict of this example with the uncertainty principle, we need to examine some numbers. Because h is so very small, 6.6  1034 J-s, the uncertainties that are implied are extremely small for macroscopic objects. If our marble has a mass of 10 g, then dividing h/m, the uncertainty principle leads to the product x v Ú 6  1034 m2/s, We can only measure the marble’s location to, at very best, the dimension of an atom, 0.5  1010 m, so that the uncertainty in speed of the marble must be at least 1022 m/s. But a velocity of this magnitude corresponds to the marble moving one atomic radius in over 15,000 years! So the uncertainty principle presents no conflict with macroscopic measurements. On the other hand, because of its small mass, to know the position of an electron to within the size of an atom implies an uncertainty in its velocity of over 107 m/s! Position and momentum are said to be conjugate variables since there is an uncertainty relation of the form of Equation (24.23) that links them together. Another important pair of conjugate variables is energy and time, with a similar minimum uncertainty relation ¢E ¢t L h.

(24.24)

This uncertainty relation has a number of significant consequences. For example, atoms in an excited state have a characteristic lifetime, the average time before emitting a photon and returning to their ground state. This is a statistical process meaning that in a large collection of such excited atoms, the average decay time (the lifetime) is a characteristic of that particular transition, but that for any particular atom undergoing this transition we cannot know the exact transition time. Because of this uncertainty in time, there is a corresponding uncertainty in the energy of the atomic transition, given by Equation (24.24) and hence in the energy of the emitted photon. We can think of this energy uncertainty as arising from a small characteristic energy width of the excited state itself. Narrower, more sharply defined, energy levels have longer lifetimes, whereas broader energy levels have shorter lifetimes. FIGURE 24.9 The uncertainty in the x-location of a particle is inversely related to the uncertainty in its momentum along the x-direction. The product of these two uncertainties must be at least the order of h  6.63  1034 J-s.

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Δx Δx Δx

Δp Δp

Δp

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Example 24.3 Find the spread in frequencies, or linewidth f, when atoms radiate from an excited state with a lifetime of 2  109 s. Also find the fractional spread in frequencies,  f/f, if the emitted photons have a wavelength of 550 nm. Solution: The lifetime of the transition leads to a spread in energy of the emitted photons. From Equation (24.24) and the fact that, from E  hf we know that E  hf, we can write that f  E/h ≈ (h/t)/h  1/t. Then because the transition time is a statistical average, its uncertainty is comparable to its value and we have that f ≈ 1/t ≈ 1/(2  109 s)  5  108 Hz. The photon frequency is given by f  c/  5.5  1014 Hz, so the fractional spread in frequencies is then 5  108/5.5  1014  9  107. This so-called “intrinsic” linewidth is usually masked by larger spreads in frequency due to thermal motions of the atoms producing random Doppler shifts in frequency.

Another consequence of this uncertainty relation is the possibility of multiphoton spectroscopy, as discussed briefly in Section 1 of Chapter 23 in connection with microscopy. To excite an atom from its ground state to an excited state requires a specific energy photon hf, corresponding to the transition energy. If the photon density is large enough so that the probability for the absorption of two or more photons within a short time t is large, then the uncertainty relation allows, for example, N photons, each of energy hf/N, to cause the overall transition even though there are no intermediate energy levels so that no transition to such intermediate energies is possible (Figure 24.10). In other words, as long as the photon absorption occurs within a very short time window, the energy uncertainty that follows from the uncertainty relation is sufficient to allow this process to occur. Tunneling, mentioned in the last section, is another type of purely quantum mechanical phenomenon that arises from the uncertainty relation. Imagine an electron confined within a one-dimensional box by potential walls, or barriers, such as the one shown in curve c of Figure 24.8, on either side of the box. Classically, if the electron had an energy less than that of the barrier height, it would forever be trapped within the box bouncing back and forth. Quantum mechanics agrees with this as well if the potential barriers are infinitely high and leads to the standing waves studied in the previous section. If the barriers are finite, however, then there is a small probability that the electron can escape or “tunnel” through the barrier wall. Tunneling can be related to the energy–time uncertainty relation. If the time for the electron to pass through the wall is short enough, then the uncertainty in the electron’s energy during that time interval may become large enough to allow its energy to exceed the barrier energy. Therefore during that brief time the electron does not violate conservation of energy and the laws of physics will not prevent the electron escaping from the box. The probability that the electron tunnels out of the box is small and depends on the barrier potential height and wall thickness. As bizarre as this appears, it is a real phenomenon and can be used in actual pieces of equipment to study materials on an atomic scale.

E2

hf = (E1 – E2 )/2

E2

hf = E1 – E2 hf = (E1 – E2)/2

E1

E1

FIGURE 24.10 (left) Absorption of a single photon causing a transition to an excited state. (right) Absorption of two photons each with half the energy needed can occur even though there is no intermediate energy level, as long as the lifetime of the (virtual) intermediate state is shorter than the minimum uncertainty dictated by the Heisenberg uncertainty principle.

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FIGURE 24.11 Scanning tunneling microscope schematic and EM image of a needle tip used for scanning.

Servo/computer

metal-coated sample

The scanning tunneling microscope uses this phenomenon to image the surface of a microscopic object with unprecedented resolution. A sample is coated with a thin layer of metal to make it electrically conducting. A fine-tipped needle is then placed close to, but not in contact with, the surface and a small potential difference is applied between the needle and the sample surface (Figure 24.11). If the tip-to-sample distance is on the order of 1 nm, then a small electric current can be detected from electrons that have tunneled across the air or vacuum insulating layer. As the needle moves along just above the surface, the gap distance changes and the tunneling current changes as well. Because the tunneling current is so sensitive to the gap (corresponding to the barrier wall thickness), extremely high resolution images of surface sample features is possible. Vertical resolution of better than 102 nm and lateral resolution about an order of magnitude less is possible, easily allowing individual small atoms at the surface to be visualized. Although, in principle the needles used should have tips with atomic dimensions, it turns out to be fairly straightforward to fabricate such needles because surfaces tend to be fairly rough on atomic dimensions anyway. One commonly used mode of operation has a feedback loop circuit to vary the height of the probe as it is scanned across the sample in order to maintain a constant height above the surface and thereby a constant sample-to-probe current. By scanning the sample, a record of the surface topography is recorded, allowing extremely high resolution of surface features (Figure 24.12).

FIGURE 24.12 False color scanning tunneling microscope images. (left) Iron atom array, produced by manipulating the atoms by the STM needle, on a copper atom support film. The wavelike appearance in the background is due to electron matter standing waves that are trapped within the iron atom “corral”; (right) native DNA image in which the stacked bases are just visible.

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4.1. QUANTUM MECHANICS AND ENERGY LEVELS

In the previous section we saw that a particle trapped in a one-dimensional box has a nonzero minimum energy, the zero-point energy, given by Equation (24.20). This agrees with the uncertainty principle, which also requires that a mass (m) confined to move in a finite region of space (of extent L) must have a smallest speed whose magnitude is approximately given by vQM ~ h/(mL). (Here we use the nonrelativistic expression for p  mv.) A confined mass, therefore, must have a minimum kinetic energy, KEQM  (1/2)m(vQM)2 ~ h2/(mL2) (remember, “~” means “order of magnitude;” “1/2” has the same order of magnitude as “1”), in qualitative agreement with Equation (24.20). This minimum, irreducible kinetic energy is also called the mass’s ground state kinetic energy. Let’s examine this in some further detail. Suppose we drop a 1 kg mass 10 cm, calculating that it acquires a KE of mgh ~ 1 J. If we substitute into our KEQM expression m  1 kg and L  10 cm (0.1 m), we get a ground state kinetic energy of about 1066 J, using h ~ 1034. Obviously, the 1 J value quoted above for a 1 kg mass falling 10 cm has nothing to do with the ground state motion of the 1 kg mass, a point we return to below. The 1 kg mass consists of about 1025 atoms. Each of these is confined to the same 10 cm as the whole body. Thus for one atom with m ~ 1025 kg and L  0.1 m, we have KEQM ~ 1041 J. If we multiply the latter kinetic energy per atom by 1025 atoms we might expect to get the kinetic energy of the whole 1 kg body. What we do get is 1016 J. Although neither the 1066 nor the 1016 values are macroscopically measurable, and, therefore, are not of much macroscopic consequence, they differ by a factor of 1050! It would be nice to know which is right. Resolution of this discrepancy revolves around the notion of coherent versus incoherent motion as discussed in Chapter 12 (see Figure 12.7). When we use 1 kg for the mass in the calculation we are tacitly assuming that all 1025 atoms in the body move together in lock-step fashion, as a coherently synchronized swarm. When we use 1025 kg for the mass we are tacitly assuming that each atom moves independently of the rest. Such unsynchronized motion is incoherent motion. In a solid, where all the atoms are glued together by interatomic forces, the former seems like a reasonable assumption. But wait! Each atom in a solid is surrounded by neighboring atoms that also confine its motion. Thus each atom shares the macroscopic confinement of the whole body, whereas at the same time each has a microscopic confinement. For an atom in a solid confined by its neighbors, m is about 1025 kg and L is about 1011 m (about 10% of the atom’s size). Thus, the ground state speed of the atom (vQM) due to this confinement is on the order of 102 m/s and its ground state kinetic energy (KEQM) is about 1021 J (you should verify these values using the equations at the beginning of this discussion above). Clearly, this motion has to be incoherent, because if all of the atoms were moving lock-step together the solid would be careening around at over 100 m/s! Because the motion is incoherent, we can add up the kinetic energies and conclude that the 1 kg solid sitting at rest has about (1021 J/atom)(1025 atoms)  104 J of ground state kinetic energy due to incoherent, microscopic atom motions (far more than the 1 J you get by dropping all of the atoms coherently a distance of 10 cm). As each atom jiggles incoherently, it carries its electrons and its nucleus with it. But the electrons are confined by their interaction with the nucleus so they have additional motion internal to the atom’s. Use the values m ~ 1030 kg and L ~ 1010 m to find that for each electron vQM ~ 106 m/s and KEQM ~ 1018 J. As there is an order of 10 e/atom in a typical solid, the incoherent motion of all electrons yields a ground state kinetic energy of about 108 J in a 1 kg mass. In addition, the nucleons in each nucleus are confined by their strong nuclear interaction with each other. For them, you should find m ~ 1027 kg and L ~ 1015–1014 m, leading to vQM ~ 107 to 108 m/s (a fair fraction of the speed of light) and KEQM ~ 1011–1013 J for each nucleon. Adding all of this kinetic energy up yields more than 1012 J in a 1 kg mass. In other words, in each macroscopic body there is a phenomenally large amount of ground state kinetic energy associated with microscopic, incoherent motion, with the overwhelming majority being associated with motion inside the atomic nuclei.

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FIGURE 24.13 Typical allowed quantum states for matter.

10–21 J 10–12 J

10–18 J }

}

Vibrational energy levels

Electronic energy levels

Nuclear energy levels

The states of motion allowed by quantum mechanics tend to have different kinetic energies. The energy differences between these allowed states tend to be about the same size as the ground state kinetic energy. Thus the allowed states of nucleon motion tend to differ in energy by about 1012 J. Electronic states tend to differ in energy by about 1018 J, and atomic vibrational states tend to differ in energy by about 1021 J (see Figure 24.13). We return to a discussion of energy levels and their study by spectroscopy in the next chapter. As we have seen in Chapter 12, the average kinetic energy of an atom or molecule is proportional to the absolute temperature, so that T ~ KEinternal  (1023 K/J), where KEinternal is an internal kinetic energy and T is measured in kelvins, K. For atomic vibrations, KEinternal is about 1021 J, so T for atomic vibrations is of order 102 K (e.g., room temperature). For electronic motion in an atom, KEinternal is about 1018 J, so T for electrons is about 105 K. For nucleonic motion in nuclei, KEinternal is about 1012 J, so T for nucleons is about 1011 K. Reciprocally, we can say that if a body has a temperature of a few 100 K it is possible to excite internal atomic vibrations, but not electronic states and, emphatically, not nucleonic states. To excite these requires very high temperatures, indeed. In a body at room temperature, all of the excess energy above the ground state is in atomic vibrations. At room temperature, the body’s electrons and nucleons are “frozen” into their respective ground states. Therefore, here’s one of the remarkable secrets of life. In a living cell (whose temperature is roughly 300 K), there’s a huge amount of nucleonic internal energy, a much less, but nonetheless significant, amount of electronic internal energy, and, by comparison, an almost negligible amount of atomic vibrational internal energy. Even so, atomic vibrations are the only energy source available for the cell to use, because the other motions are stuck in their ground states. By carefully marshalling and partitioning its puny supply of internal energy, a cell manages to perform all the various tasks of life, including protein replication, locomotion, and cell division.

CHAPTER SUMMARY The theory of special relativity is based on two fundamental postulates: all the laws of physics are the same in all inertial frames of reference and the speed of light in vacuum has the same value c in all inertial reference frames. This latter postulate seems contrary to our (lowspeed) intuition and leads to a large variety of seemingly bizarre, but experimentally confirmed, effects

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having to do with time and space. Here we focus on the dynamical quantities that we need in the next chapters. Momentum of a particle of mass m moving at a velocity v is given by p

mv

11  v 2 /c2

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 gmv,

AND

(24.1)

QUA N T U M P H Y S I C S

effect also treats high-energy x-ray photons as particles colliding with electrons to successfully analyze the scattering results. There is also a further discussion in the chapter of the double-slit interference experiment, but now for single photons, or for electrons, which have a deBroglie wavelength given by

where g

1

11  v2 /c2

.

Similarly, the particle’s relativistic energy is given by E

mc2

11  v2 /c2

 gmc2,

(24.5)

which can also be written as the sum of the kinetic energy KE and the rest energy, as E  KE  mc2.

(24.6)

Small changes in the (rest) mass m, lead to large changes in energy given by ¢E  ¢mc2,

(24.7)

and this effect has led, for example, to both atomic bombs and nuclear power plants. Energy and momentum are connected through the equation E2  p2c2  m2c4.

(24.8)

For a massless particle, such as the photon, or in the limit that the velocity approaches c (so that  becomes very large) this last equation reduces to E  pc.

(if m  0 or  >> 1).

(24.9)

Historically, several important experiments revealed the “particlelike” nature of photons and initiated the notion of “wave–particle duality,” the idea that all elementary particles exhibit both wave- and particlelike properties depending on their interactions. The photoelectric effect is the production of an electric current proportional to the incident light intensity. But, each incident photon of frequency f needs a minimum threshold energy, the work function , in order to liberate an electron, and because E  hf for a photon, with h  Planck’s constant  6.63  1034 J-s, there will also be a minimum frequency needed. Einstein worked out the explanation for this effect and found that the liberated electrons have a maximum KE given by KEmax  hf  £,

(24.13)

obtained simply from conservation of energy in the individual photon–electron interaction. The Compton

C H A P T E R S U M M A RY

h l . p This association of a wavelength with a “particlelike” momentum bridges the wave–particle duality notion. Electrons can be seen to exhibit wavelike properties in the phenomenon of electron diffraction, for example. Quantum mechanics, the theory of the microscopic world, has a central dogma that all the possible information knowable about a system, for example, an electron, can be described by the wave function , whose square is given by ° 2 (x, y, z, t)¢V  Probability to find electron within ¢V at (x,y,z) at time t. (24.16) The wave function can be found by solving the Schrödinger equation, which leads to a set of quantum numbers that define the possible energy levels, angular momentum, spin, and so on of the system. A fundamental rule is the Pauli exclusion principle, which states that no two interacting fundamental particles (e.g., electrons) can have the same set of quantum numbers. Another fundamental principle is the Heisenberg uncertainty principle, which describes a basic limitation in nature on the simultaneous measurement of pairs of conjugate variables ¢x ¢p L h,

(24.23)

¢E ¢t L h.

(24.24)

These limitations have negligible effect in the macroscopic world, but can produce major effects in the microscopic arena. One notable result is the phenomenon of tunneling and an associated microscopy technique known as scanning tunneling microscopy, which gives atomic resolution images. A quantum mechanical analysis of the ground state energy contained in matter shows that internal KE from incoherent electron and nucleon motions is huge, much larger than typical translational energies of matter. The different internal energies have corresponding energy levels that can be explored with spectroscopy.

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QUESTIONS 1. Relativity requires any particle that travels at the speed of light, such as the photon, to have no rest mass. Why is this necessary? 2. In a photoelectric effect experiment, if a beam of green light produces a photocurrent, will a beam of blue light with the same intensity produce a larger, smaller, or the same photocurrent? 3. If one shade of yellow photons will produce a photocurrent, but another shade does not, will red light produce any photocurrent? Will green light? 4. If when the anode voltage is set to 1.5 V there is just no photocurrent with a particular green wavelength of light, when the wavelength is changed to a blue and the intensity of the blue light is 1/2 that of the green, what happens to the photocurrent? To the maximum kinetic energy of the photoelectrons? 5. Note that the Compton wavelength shift is independent of the actual wavelength of the x-ray photon. How does the percent change in the wavelength of a Compton scattered x-ray photon depend on the wavelength of the photon? 6. How does the Compton wavelength shift for x-ray scattering from protons compare to that from electrons? 7. Discuss how the interference pattern observed in a double-slit experiment with electrons depends on the energy of the electrons. 8. For a particle in a one-dimensional box of length L, where is the particle most likely to be found when in the ground state? In the first excited state? 9. Why is it impossible for an object to be exactly at rest? Discuss this in connection with a car at a stoplight and with an atom in an “atom trap”. What is the approximate uncertainty in velocity in each of these cases? 10. Classical physics only allows a photon to be absorbed by a sample if it has an energy equal to the energy difference between the final state and the initial state. If there are no intermediate energy levels between these then no photons with less energy can be absorbed. On the other hand, experimentally it is found that if three photons from a high-intensity laser, each having an energy equal to 1/3 that of that energy difference are absorbed, the sample can reach the final state. Discuss the energy–time uncertainty relation’s impact on allowing this process to occur. 11. How do you expect the electron tunneling current to depend on the barrier height? On the barrier thickness? On the electron energy? 12. What is the advantage of false color in representing image data? Think of your nightly weather Doppler radar images. MULTIPLE CHOICE QUESTIONS 1. As a particle’s speed approaches the speed of light, its energy (a) approaches mc2, where m is the rest mass, (b) approaches its kinetic energy, 1/2 mv2, (c) approaches

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the product of the particles momentum and the speed of light, pc, (d) approaches m, where  is the Lorentz factor. 2. Which of the following is not true about the photoelectric effect? In each case assume that light of a given color is directed onto the emitter plate and a current of electrons is observed to be ejected from the plate. (a) The maximum kinetic energy of the ejected electrons is independent of the intensity of the light. (b) When the intensity of the light is lowered below a finite critical value that depends on the material of the emitter plate, the current abruptly stops. (c) It takes the same very short time to produce a current after turning the light on when the light has intensity I as when it is has intensity I/2. (d) The work function of the emitter plate is independent of the color of the light. Questions 3 and 4 refer to an intensity I of yellow light incident on an ideal 100% efficient metal emitter surface in a phototube producing photoelectrons at a rate of N photons per second. 3. Shining blue light of intensity I/2 on the same metal surface in the phototube will (a) not produce any photons, (b) produce 2N photons/s, (c) produce N/2 photons/s, (d) it is impossible to predict the outcome. 4. Shining red light of intensity 2I on the same metal surface in the phototube will (a) produce 2N photons/s, (b) produce N/2 photons/s, (c) not produce any photons, (d) it is impossible to predict the outcome. 5. The de Broglie wavelength of an electron is associated with what kind of wave? (a) Electric field, (b) magnetic field, (c) probability, (d) sound. 6. The ratio of the Compton shift at forward scattering to that at backward scattering is (a) 2, (b) 1, (c) 0, (d) 1/2. Questions 7–9 refer to the particle in a box problem, where the particle is confined between 0 and L. 7. A particle in its first excited state is most likely to be found at (a) L/2, (b) L/3, (c) L/4, (d) L. 8. A particle in its second excited state will never be found at (a) L/4, (b) L/3, (c) L/2, (d) it can be found everywhere in the box at some time. 9. When a particle in a box makes a transition from its third excited state to its ground state, the emitted energy equals (a) 9, (b) 5, (c) 8, (d) 2 times its zeropoint energy. 10. In quantum mechanics an electron is viewed as being described by a wave function. When confined to a finite region of space, the allowed electron wave functions are standing waves. This explains (a) the results of the photoelectric effect, (b) the results of Compton scattering, (c) why an atom must have a lowest energy state in which its electrons cannot radiate away energy, (d) why the sky is blue. 11. The principle of complementarity refers most closely to (a) the uncertainty principle, (b) wave–particle duality, (c) tunneling, (d) zero-point energy.

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12. Heisenberg’s uncertainty principle (a) only applies to atomic and subatomic particles, (b) predicts large uncertainties in the velocities of macroscopic objects at rest, (c) states that the product of uncertainties in conjugate variables cannot be zero, (d) explains experimental uncertainties in all measured quantities. 13. Tunneling refers to all but which of the following? (a) An electron escaping from a potential well, (b) an electron traveling in a classically inaccessible region of space for a short time, (c) an electron traveling down a channel between atoms, or a tunnel, in a material, (d) the process used in the STM to image atoms. 14. A scanning tunneling microscope requires all but the following: (a) a fine-tipped needle, (b) a stable, vibration-free sample holder, (c) a vacuum pump to put the sample under vacuum, (d) a stable micromotor to move the needle or sample about. 15. The ground state kinetic energy of a macroscopic body consists mostly of (a) coherent motion of the body as a whole, (b) incoherent motion of the electrons of the atoms, (c) incoherent motions of the nucleons, (d) coherent motions of the atoms of the material. 16. When a block slides across a rough horizontal table surface and stops (a) its coherent center-of-mass energy is transformed into internal kinetic energy, (b) its incoherent atomic energy is transformed into incoherent nucleon energy, (c) its coherent center-ofmass energy is transformed into coherent atomic energy, (d) its coherent center-of-mass energy is transformed into photon energy. 17. In order of increasing energy, the different types of energy of a macroscopic body are due to (a) incoherent nucleon motion, incoherent electron motion, incoherent atomic vibrational motion, coherent center-of-mass motion, (b) coherent center-of-mass motion, incoherent atomic vibrational motion, incoherent electron motion, incoherent nucleon motion, (c) incoherent atomic vibrational motion, incoherent electron motion, incoherent nucleon motion, coherent center-of-mass motion, (d) coherent center-of-mass motion, incoherent nucleon motion, incoherent electron motion, incoherent atomic vibrational motion. PROBLEMS 1. Compute the momentum and energy of a 1 kg rest mass object traveling at v  0.8 c, 0.9 c, 0.95 c, 0.99 c, and 0.999 c. 2. Repeat the previous problem for an electron and calculate the energy in MeV. 3. Fill in the steps in the derivation of the classical limit of Equation (24.2). 4. Derive Equation (24.8), the connection between relativistic energy and momentum.

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5. Calculate the energy of each of the two photons produced from electron–positron pair annihilation when the electron and positron were nearly at rest. What is their wavelength? 6. A 2.5 MeV photon passes near a stationary atom and produces an electron–positron pair. If all the energy of the photon goes into creating the pair, what is the speed of each when produced? 7. What are the momentum, wavelength, and frequency of a 1.2 MeV photon traveling in space? 8. The work function for cesium is 2.9 eV. Suppose a vacuum tube with a cesium photocathode is configured for the photoelectric effect. (a) What is the maximum wavelength photon that will produce a photocurrent? (b) If 400 nm photons are used what is the maximum kinetic energy of the emitted electrons? (c) If a 1 W beam of 400 nm photons is used, what is the photocurrent that will be detected assuming 100% efficiency (i.e., assuming all emitted photoelectrons are collected by the anode)? (d) What maximum work function is needed to allow photoelectron emission using green photons of 500 nm wavelength? 9. Photons of 400 nm wavelength are incident on a photocathode. As the anode potential is made more negative, the photocurrent decreases until it reaches zero when the anode voltage is 0.82 V. Find the work function of the photocathode. 10. A photoelectric experiment is conducted with a sodium surface with work function  2.28 eV. (a) When the surface is illuminated with light with a wavelength of 410 nm, what are the speed and kinetic energy of the emitted electron? (b) Is the electron relativistic? (c) What is the minimum frequency needed to detect a photocurrent? (d) What is the maximum wavelength of light that can be used to detect a photocurrent? (e) What are the speed and kinetic energy of the emitted electron if the incident light is 700 nm on the same sodium surface? 11. Suppose that 134Cs, a gamma ray emitter, is used in a Compton effect experiment and the gamma rays are observed to scatter from electrons in an Al target at a 50° angle. 134Cs is radioactive and decays by producing a 1.6 MeV gamma ray, which is just like an x-ray except it has a higher energy. (134Cs also emits  particles in addition to  rays and has a half-life of about 2.1 years, both of which have nothing to do with the problem.) (a) What is the wavelength and momentum of the incident gamma ray? (b) Write an expression for the energy of the scattered photon as a function of incident energy photon and the scattering angle .

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12.

13. 14.

15.

16.

(c) What is the energy of the scattered -ray photon in MeV? (d) What is the kinetic energy (in MeV) of the recoiling electron? (e) What is the speed of the recoiling electron as a fraction of c? A 0.012 nm wavelength beam of x-rays is incident on a foil target. (a) What is the incident x-ray photon energy in MeV? (b) What is the wavelength and energy of backscattered Compton x-rays? (c) How much energy is given to the foil target for each backscattered x-ray? Find the relativistic energy (in MeV) of an electron with a de Broglie wavelength of 0.0012 nm. After learning about de Broglie’s hypothesis that particles of momentum p have wave characteristics with wavelength   h/p, a 65 kg student has grown concerned about being diffracted when passing through a 90 cm wide doorway. (a) If the student is traveling at a whopping 0.5 m/s, what is the student’s momentum? (b) What is the de Broglie wavelength of the student? (c) What would the size of the door need to be in order for there to be noticeable diffraction of the student? Suppose that a 1 mW He–Ne laser (  633 nm) shines on a screen. How many photons strike the screen each second? (No wonder we are not aware of individual photons!) An electron is trapped in a 10 nm one-dimensional deep potential well. Find the following. (a) Its ground state energy

602

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(b) The energy of the second excited state above the ground state (c) The minimum quantum number n corresponding to an energy of at least 100 eV Show that for a particle in a box the difference in energy between consecutive energy levels increases in proportion to the quantum number n. An atomic transition from an excited state to the ground state has a lifetime of 108 s. What is the uncertainty in the energy of the approximately 550 nm photon emitted? What is the uncertainty in the wavelength of the photon? What is the minimum velocity of an electron in a hydrogen atom, confined within a distance of about 0.1 nm? What is the minimum uncertainty in the velocity of a 2000 kg truck waiting at a red light (or its maximum possible velocity) when its position is measured to an uncertainty of 1.0  1010 m. An electron travels down a channel between two parallel arrays of large atoms along the x-axis separated by 0.12 nm. What is the minimum uncertainty in the y-momentum of the electron? Using the uncertainty principle, derive Equation (24.19) for the zero-point energy of a particle in a box, apart from a small numerical factor. Alpha decay in radioactive nuclei can be thought of as the escape of a helium nucleus from the attractive barrier potential of the larger nucleus. If the nucleus diameter is 5.5 fm, find the maximum velocity of the alpha particle in the nucleus.

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