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Special Relativity and
Quantum Physics
24
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 to find that
KE mc2 a1 v2
1
b mc2 mv2.
2
2
2c
(24.4)
Therefore, as long as v/c << 1, we see that the relativistic kinetic energy reduces to
our usual classical physics expression. The relativistic expression for kinetic energy
also confirms the idea that it becomes more and more difficult to accelerate a particle of mass m as its speed approaches c because the kinetic energy also grows very
rapidly, in proportion to .
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Now we make a leap in our interpretation of Equation (24.2).
We define the total relativistic energy of the particle to be the first
term in Equation (24.2)
E
mc2
11
v2 /c2
gmc2.
(24.5)
Then, from Equation (24.2), we can rewrite this as
E KE mc2.
(24.6)
The total relativistic energy of a particle is therefore made up of its
kinetic energy, the first term, and its rest energy, given by mc2, the energy
remaining when v 0 or 1.
Example 24.1 For an electron traveling at v 0.95 c, find its momentum, total
energy, and kinetic energy and express each of these as multiples of their classical
(nonrelativistic) values.
Solution: An electron has a rest mass of 9.1 1031 kg and, at v 0.95 c,
1
3.2.
a value of g 11 (0.95)2
Therefore its momentum is equal to p mv (3.2)(9.1 1031) (0.95)
(3 108) 8.2 1022 kg-m/s, its total energy is equal to mc2 2.6 1013 J, and its kinetic energy is E mc2 1.8 1013 J. Because the classical momentum is just mv, the relativistic momentum is exactly a factor of
3.2 larger. Classically the total energy and kinetic energy are both equal
(because there are no potential energies for an isolated electron) and equal to
1
2
14 J, and thus the total energy and kinetic energy are actually
2 mv 3.710
larger than this by factors of 7.0 and 4.9, respectively.
Einstein’s famous formula E mc2 is really a relation between the rest energy
and mass of a particle and shows the equivalence of mass and energy. A particle and
its antiparticle, for example, an electron and a positron, each with the same mass, can
annihilate on collision converting all of their mass to pure energy in the form of photons (as long as all conservation laws are satisfied). In the inverse reaction, known as
pair production, a gamma ray photon with enough energy can create an electron and
positron pair. In this case if the photon has an energy greater than the combined rest
mass of the electron and positron, then these two particles share the remaining energy
in the form of kinetic energy as they fly apart at some appropriate speed.
Nuclear reactions involve small changes in the mass of nuclei with accompanying large changes in energy given by
¢E ¢mc2.
(24.7)
For example, a mass change of 1 kg leads to an energy release of about 9 1016 J, or enough energy for a U.S. city of 12 million people for a year. Even chemical reactions involve small mass changes of the reacting atoms, although the equivalent energies are much smaller than those of nuclear reactions.
Another way to consider Equations (24.1) and (24.5) for the relativistic momentum and energy of a particle is to consider the term
m
gm
11 v2 /c2
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as a variable, known as the relativistic mass (with m known as the rest mass), that
depends on the speed of the particle. Viewed in this way, the (relativistic) mass of a
particle increases dramatically with speed. This provides an alternative explanation
of why it is impossible to surpass the speed of light. The faster a particle travels, the
more massive it becomes and the more difficult it becomes to keep it accelerating.
Because the relativistic mass grows, unbounded, as the speed approaches c, no finite
force can accelerate the material object to the speed of light.
We conclude this section by showing the connection between energy and
momentum. Classically, kinetic energy and momentum are related by KE p2/2m.
With some algebra (see Problem 4), we can show that the relativistic momentum and
energy are related by
E2 p2c2 m2c4.
(24.8)
For a massless particle, such as a photon or neutrino, the rest energy term vanishes
and the energy and momentum are proportional to each other
E pc.
(if m 0 or >> 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
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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.
W AV E F U N C T I O N S ; T H E S C H R Ö D I N G E R E Q U AT I O N
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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
S P E C I A L R E L AT I V I T Y
AND
QUA N T U M P H Y S I C S
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
U N C E R TA I N T Y P R I N C I P L E ; S C A N N I N G T U N N E L I N G M I C R O S C O P E
(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
S P E C I A L R E L AT I V I T Y
AND
QUA N T U M P H Y S I C S
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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595
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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S P E C I A L R E L AT I V I T Y
AND
QUA N T U M P H Y S I C S
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
598
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
S P E C I A L R E L AT I V I T Y
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.
S P E C I A L R E L AT I V I T Y
AND
QUA N T U M P H Y S I C S
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.
QU E S T I O N S / P RO B L E M S
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
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18.
19.
20.
21.
22.
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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.
S P E C I A L R E L AT I V I T Y
AND
QUA N T U M P H Y S I C S