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CHAPTER I
Review of Modern Physics
A. Review of Important Experiments
Quantum Mechanics is analogous to Newtonian Mechanics in that it is basically a system
of “rules” which describe what happens at the atomic level. Much like the development of
relativity, where high speeds give rise to a “new” picture of nature, the quantum mechanical
description of nature was developed as a result of several experiments which could not be
properly explained using the classical description of matter. It was found that, at the atomic
level, a number of physical quantities were quantized rather than continuous. The most
important experiments and the significance of each are listed below as review.
Blackbody Radiation: Max Planck was able to successfully derive the correct expression
for the energy density per frequency interval of a blackbody radiator of temperature X
? / ß X . / œ
)1 2
/$
./
- $ /2/ Î5X "
(1.1)
by assuming that the energy of the oscillators which make up the wall of a blackbody radiator
must be quantized, rather than continuous, i.e., that the radiators have energy given by
I œ 82/ , where / is the frequency of oscillation of the oscillator, 8 is an integer, and 2 (now
called Planck's constant) is a very small constant ( ¸ 10-$% J-s). Thus, the energy inside the
cavity could not take on any arbitrary value of energy but rather must be thought of as coming
in “chunks” whose size was demonstrated to be given by the equation:
I œ 2/
(1.2)
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Problem 1.1 The energy which is emitted from a heated cavity in a frequency range between
/ and / . / per second, per cross-sectional area of cavity opening is given by
R/ 2/
8/ 2 /
./ œ
- . / œ ?Ð/ Ñ . /
?E ?>
%
%
where R/ . / is the number of photons in the energy interval between / and / . / , and 8/ is
the number density of photons in this frequency interval, 2/ is the energy of a given photon, is the speed of a photon, ?Ð/ Ñ . / is the energy density within the cavity, and ?E and ?> are
the area of the opening, and the time interval, respectively (in the limit as these two go to
zero).
Using the spherical geometry designated in the diagram below, derive this last equation
by assuming that the photons in the blackbody cavity are distributed uniformly throughout the
volume and that they are emitted isotropically from any arbitrary volume .Z . Notice that only
those photons which are directed toward the opening (designated by .E) will escape the
cavity. Also notice that only those photons which are within a distance ?< œ - ?> can escape
the cavity in a time ?>.
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dV
θ
dΩ
r
dA
Hint: First, determine the number of photons in the volume .Z that are moving toward the
opening of area .E. Then express the volume .Z in sperical coordinates, changing the
differential .< to - .>, and then integrate over all values of ) and 9. Each photon has energy
2/, so you can determine the energy carried through the cavity opening per unit time per unit
area of opening.
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Problem 1.2 The energy density per unit frequency interval for a blackbody radiator is given
by
? / ß X . / œ
)1 2
/$
./
- $ /2/ Î5X "
Express this equation in terms of the energy density per unit wavelength interval under
the assumption that ?/ ß X . / œ ?-ß X . -Þ
(b) Use the expression you derive for ?-ß X to determine the wavelengthß -7+B ß for which
the energy density is maximizedÞ Show that your expression has the form -7+B X œ ,,
where , is a constant. Calculate the value of the constant , and determine the value of
-7+B for the Sun where the surface temperature is taken to be 6000K. [Hint: To evaluate
the constant , you will need to solve the equation & B œ &/
B . You can do this
using the function 0 D/<9 in MatLab.]
(c) Determine the frequencyß /7+B ß for which the energy density is maximized, and show
that this frequency is not the frequency that you might expect from the equation -/ œ - .
In fact, the wavelength associated with /7+B is about twice the value of -7+B . This
means that for the sun (with a surface temperature of about 6000K) -7+B occurs in the
visible, while the wavelength associated with /7+B occurs in the infrared! Can you
explain why this occurs? [Hint: How is the size of . / related to the size of . -?]
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(a)
The Photoelectric Effect: Einstein carried the idea of quantized energy further by
assuming that the energy of the electromagnetic waves was quantized. Using this hypothesis,
he was able to derive Planck's equation for blackbody radiation from an alternate approach.
The postulate that light could loose energy only in “chunks” (called photons) of energy 2/ also
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explained the observed characteristics of the photoelectric effect, that the kinetic energy of
photoelectrons is given by
"
7@# œ 2/ /9
#
(1.3)
where "# 7@# is the kinetic energy of the emitted electron, 2/ is the energy of the incident
photon, and /9 is the work function of the metal.
This explanation of the photoelectric effect set the stage for a new view of the nature of
light. The interference experiments of Young had clearly established that light had wave
properties (all of physical optics is based upon this observation). This new view of light
seemed to demand a particle-like description of light as well.
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Problem 1.3 The maximum kinetic energy of photoelectrons ejected from an aluminum target
is measured to be 2.3 eV for radiation of 200 nm and 0.90 eV for radiation of 258 nm. Use
this data to calculate Planck's constant and the work function for aluminum.
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Bohr's Theory of the Atom: Bohr applied the idea of the quantized nature of light in his
treatment of the hydrogen atom. He showed that a proper description of the hydrogen atom
depended upon the existence of quantized energy levels within the atom and the assumption
that no energy was radiated while an electron was in one of these quantized energy levels
(called bound states or stationary states). He postulated that radiation occurred only when an
electron moved from one quantized energy level to another, giving rise to the quantized nature
of the atomic spectra:
2/ œ 2-Î- œ I9 ”
"
"
•
7#
8#
(1.4)
where I‰ is 13.6 eV, and 7 and 8 are integers.
Bohr's theory led to two very important conclusions À 1) When the motion of a particle is
restricted in some fashion, the energies associated with that particle are quantized in
particular, he demonstrated that the angular momentum of the hydrogen atom was quantized.
2) The quantum nature of matter must be consistent with classical physics when the quantum
numbers become large. This is the so-called correspondence principle (see Appendix 1.A).
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Problem 1.4 Bohr postulated the quantization of the angular momentum, i.e., that P œ 8h .
With this assumption, he was able to generate the equation given above that precisely predicts
the spectral lines that arise from transitions within the hydrogen atom. Show that the
requirement of the quantization of angular momentum can be satisfied if we assume that an
electron orbiting a proton can be described by a wave moving in a circular path around the
proton, and that this wave must interfer constructively with itself.
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The Compton Effect: It was deBroglie's work, however, that really indicated the need for
a new view of the atomic nature of matter. According to Einstein's theory of special relativity,
the total energy of a particle is given by
I # œ : # - # ˆ7 o - # ‰
#
(1.5)
Now, if we assume that the photon is a particle of zero rest mass (which is required if the
photon is to move at the speed of light) this equation reduces to
I œ :-
(1.6)
and, using the Einstein relation I œ 2/ , we have
I œ 2/ œ :- Ê : œ 2/ Î- œ 2Î-
(1.7)
The equation : œ 2Î- is called the deBroglie equation, and provides the connection between
the wave nature of light described by the wavelength - and the particle nature of light
described by the momentum :. The so-called wave-particle duality of light, therefore, is
embodied in this equation.
Compton's successful calculation of the change in wavelength of x-rays scattered by
electrons in a solid was based upon this equation. He assumed that an x-ray “photon” could be
treated like a particle with energy 2/ and with momentum : œ 2Î-. He then used
conservation of the relativistic energy and momentum of the incident photon and the target
electron to derive the so-called Compton Equation:
-w - œ
2
" -9= )
7/ -
(1.8)
B. The Wave-Particle Duality of Nature
The Compton effect, the photoelectric effect, and the measured blackbody radiation
spectrum clearly demonstrated the “particle nature” of radiation in contrast to the previously
accepted and demonstrated “wave nature” of radiation. This led to a wave-particle duality,
i.e., a dual description of the nature of electromagnetic radiation. In some cases it “acts” like a
wave and in some cases it “acts” like a particle! We will discuss this further in a moment.
However, deBroglie pushed the idea of the wave-particle duality even further. He argued
that just as light (previously thought of as a wave) could be thought of as a particle, particles
might also be thought of as waves. He based this argument upon the observation that light can
be treated in two different ways as “waves” in physical optics, and as “rays” in geometrical
optics. The method used to treat light depends upon the relative size of the wavelength of light
and the object with which the light is interacting. When the wavelength of light is small
relative to the size of the objects with which it is interacting we can use geometrical optics and
“ray-tracing” techniques (where we treat light as if it moves in straight lines, but is “bent” at
interfaces). When the wavelength of light is of the same size as the particles with which it is
interacting, however, light must be treated as a wave. The classical picture of particles moving
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is straight lines, deBroglie argued, might be analogous to “ray-tracing” in optics. Thus,
particles such as electrons and protons would be expected to exhibit wave-like characteristics,
since the wavelength of such particles is comparable to the size of atomic systems. The
wavelength of classical particles, however, is typically much smaller that the size of a nucleus
and therefore shows no “diffraction” effects, but travels in straight lines.
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Problem 1.5 This problem helps you get a feel for the wavelength of various objects.
a) Calculate the wavlength of a baseball thrown at 90 miles per hour. Assume that the
baseball has a mass of 145 grams.
b) Calculate the wavelength of an electron accelerated through a potential difference of 1
keV.
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Davisson-Germer Experiment: This validity of deBroglie's hypothesis was demonstrated
by Davisson and Germer who observed the diffraction of an electron beam incident upon a
nickel crystal [see Fig. 1.2 in Ohanian]. Davisson and Germer were able to describe their
results based upon the assumption that the electrons had a “wavelength” associated with them
which was given by
- œ 2Î:
(1.9)
:#
œ /Z
#7
(1.10)
where
C. Consequences of Wave-Particle Duality
The wave-particle duality as encountered in electromagnetic phenomena and also the same
duality as expressed by deBroglie's hypothesis leads to some interesting philosophical
questions: 1) Are photons particles or waves? What about electrons? 2) How can a single
entity have both particle- and wave-like characteristics?
We will consider a few examples which will help in developing a consistent picture of this
wave-particle duality.
1. Young's interference experiments. When light is incident upon an opaque screen
having two slits, separated by a distance d, there arises behind the screen an interference
pattern such that intensity maxima occur where
7- œ . sin)
7 œ "ß #ß $ß á
(1.11)
This interference pattern is a result of the wave-nature of light and the fact that the resultant
wave function at any point is the sum to all incident waves at that point, and that the intensity
of the light at a point is equal to the square of the resultant amplitude. All this arises from the
fact that the classical wave equation is a linear equation and that the sum of any two solutions
to the wave equation is also a solution to the same wave equation. But from the point of view
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of photons, we wonder if this “interference” is between different photons, or is it more
fundamental?
If we consider a very weak light source, it is conceivable that only one “photon” will
reach the screen at any one instant of time. We can demonstrate the “quantum” nature of the
photons in the light beam by looking at a photographic emulsion placed behind the opaque
screen. We will see that single points on the emulsion are being “activated”. That is, the
interference pattern which is observed is not being produced at all points on the screen at the
same time, but rather single points are darkened one at a time due to the interaction of the
photon with the emulsion, and the location of individual points appears almost random.
However, given enough time, the single spots are found to be grouped such that a large
number of darkened spots occur in the region where a maximum intensity of light occurs,
while only a few dark spots occur in regions of low intensity. Eventually these regions begin
to look like a continuous variation in the darkening of the emulsion - thus exhibiting what
looks like a continuous interference pattern.
Now some interesting questions arise. What if we closed one of the slits? Obviously we
would no longer have an interference pattern, but a diffraction pattern. How do the photons
know how many slits are open? Are they somehow spread out in space so that they sense the
presence of another slit? Perhaps, as the photons approach the slits they might somehow
interact with one another and thus give some sort of uncertainty as to the position of the
photons as they reach the slits, and that the presence of two slits gives rise to the interference
pattern. However, if the intensity of light is low enough so that only one photon approaches the
opaque screen at a time, we still get the same results!
2. Light polarization. Consider light incident upon two polarizers. The first polarizer
acts to give linearly polarized light. The second polarizer is usually called an analyzer. If the
axis of the second polarizer is parallel to that of the first, the light passes through unattenuated.
However, if the axis of the analyzer is inclined to that of the first polarizer by an angle ), the
intensity of light passing through the system is reduced. The electric field vector It" which
describes the field after passing through the first polarizer is modified by the second polarizer
according to the equation
It # œ It " -9= )
(1.12)
The intensity of the electromagnetic wave, which is related to the square of the electric field
#
vector, is thus proportional to It " -9=# ), or M" -9=# ). However, we must take an entirely
different point of view if we consider light to be made up of photons. The photon does not
“partially” go through the polarizer. The energy of a photon is 2/ and cannot be subdivided.
Thus the photon must pass through or not pass through. We are therefore led to the idea that
we must be talking about the probability that a given photon may pass through the analyzer.
3. Light emitted from a star. In the same way, when light is emitted from a single point
source, such as a star (if we are far enough away from it), we treat this mathematically by
introducing a spherical wave propagating outward from the star. If we are located very far
away from a star we may detect the light as single photons. Does this mean that the photon is
somehow spread over all of space as it travels outward from the star, and then somehow
collapses into a single photon at a point where someone is trying to detect the light? It would
seem that the best explanation would be that the mathematical model describes the likelihood
of finding a particle; i.e., that all directions are equally likely, and that given enough photons at
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the source, we might expect to find an equal number of photons distributed in all angular
directions throughout space.
Thus we are led to the conceptual picture first solidly proposed by Max Born in which we
associate the “particles” of nature (photons, electrons, protons, etc.) with a probability wave
which simply indicates the likelihood of finding a particle at a given location in space and
time.
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Problem 1.6 The classical one-dimensional wave equation
` # GÐBß >Ñ
" ` # GÐBß >Ñ
œ!
`B#
-#
`>#
is a linear second-order differential equation in position and time. Show that if 9" Bß > is a
solution to this equation, and if 9# Bß > is also a solution to this equation, then the sum
GÐBß >Ñ œ +" 9" Bß > +# 9# Bß >
is a solution to this same differential equation for any arbitrary values of +" and +# .
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D. Relationship between Photons and the Electric Field Vector
We would like to develop a more rigorous mathematical connection between the particle
and wave nature of light and particles. To do this, we first examine what we have already
learned about light. Light can be described by a wave equation, but we know that the energy
of light is given by the equation I œ 2/ . We will begin by examining the energy content of a
light beam as described by classical waves, and compare this with the “quantum” picture
which treats the beam as being made up of photons.
From electromagnetic theory, we can show that
t † Wt ‹.@ œ ` ( Œ %9 I  Œ .9 L  .@
t .+ œ ( Šf
* Wt † 8
Ÿ
`>  #
#
#
#
(1.13)
where the right-hand-side is the time derivative of the energy of the electromagnetic field
inside of a volume Z . The negative sign indicates that we are dealing with a decrease in
energy within the volume Z . Associated with this decrease in energy is the integral over the
surface E (surrounding the enclosed volume Z ) of the Poynting Vector Wt (<t,>). Thus, Wt is
associated with the flow of energy out of an enclosed volume, so that Wt is the rate at which
energy flows outward per unit area at a point <t on the surface E , which encloses the volume
Z . Now the electric and magnetic fields change in time, so that we must use time averages
and calculate the time averaged Poynting Vector to get any significant meaning. We state,
without proof, that the time averaged Poynting vector can be written in terms of either the
electric or the magnetic field (since they are related to one another), in the form
t œ " dÐI
t ‚ Lt ‡ Ñ œ " %9 -lIÐ<ß
t >ÑI ‡ Ð<ß
t >Ñl s
ØWÙ
5
#
#
(1.14)
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where we see that the time averaged flow of energy from an enclosed volume is associated
with the absolute magnitude squared of the electric field vector (the solution to the wave
equation).
Now if we consider the “particle” picture of light, we can calculate the rate at which
energy leaves an enclosed volume. We showed in problem 1 that this must be given by
t œ " Ø8Ù ‚ 2/ ‚ ØWÙ
%
(1.15)
where Ø8Ù is the average number density of photons within an enclosed chamber, 2/ is the
energy of a single photon, and c is the speed of light. Comparing these last two equations, we
conclude that the average number density of photons within the enclosed volume must be
related to the absolute magnitude squared of the electric field vector (the solution to the
electromagnetic wave equation) within this enclosed volume, or:
Ø8Ù º ÒIÐ<ß >ÑI ‡ Ð<ß >ÑÓ
(1.16)
This is an extremely important result. Simply stated, the solution to the electromagnetic wave
equation can be related to the particle aspect of electromagnetic radiation through the square
of the absolute magnitude of the wave function! This idea and Bohr's correspondence principle
form a fundamental basis for the understanding and development of quantum wave mechanics.
As we have seen, the de Broglie - Einstein equations, and the corresponding waveparticle duality are applicable to all particles, but the observation of quantum phenomena,
occur only when the wavelength of the particle is comparable in size to something the particle
interacts with. For a single particle, we postulate that the wave equation describes some sort
of “probability” wave that is associated with the probability of finding the particle at a certain
spot at a certain time. Thus, if we can determine a solution to Schrödinger's wave equation,
the probability is associated with the square of the absolute magnitude of the solution to this
equation. For this wavefunction to successfully describe the probable motion of a particle
through space, it will have to satisfy certain criteria which we will discuss in the next two
chapters.
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NOTE: The Continuity Equation (Eq. 1.13) is an equation of the form
t † t4‹.@ œ ` ( 3 .@
t .+ œ ( Šf
* t4 † 8
`>
This equation basically states that there is a current density t4 associated with a
decrease in the charge density found inside an enclosed volume. It is basically a
conservation law. Thus, for a substance which cannot be created or destroyed, if
the amount of the “stuff” decreases within a volume, there must be a flow of the
“stuff” out through the enclosing surface! The equality of the first two integrals is
a result of the divergence theorem (Gauss's law). Since the volume under
consideration is the same for the last two terms, we can equate the integrands to
obtain the differential form of the continuity equation:
t † t4 ` 3 œ ! .
f
`>
This equation will play an important role in our future development of quantum
theory.