Module 6
Chapters 4-6
A. The Atomic Models of Thomson and Rutherford
By the end of the nineteenth century most physicists and chemists (with a few
notable exceptions) believed in an atomic theory of matter, even though no one had ever
observed an atom directly. Origins of atomic theory date back to the Greek philosophers,
who imagined atoms to be featureless hard spheres. Even though scientists of the late
nineteenth century did not have technology to see things as small as atoms, they believed
that atoms were composite structures having an internal structure. There are some
similarities between how physicists addressed their atomic theories in the late nineteenth
century and how elementary particle physicists still search for the underlying structure of
the building blocks of matter.
We mention three pieces of evidence that physicists and chemists had in 1900 to
indicate that the atom was not a fundamental unit. First, there seemed to be too many
kinds of atoms, each belonging to a distinct chemical element. The original Greek idea
was that there were four types of atoms—earth, air, water, and fire—which combined to
make the various kinds of matter we observe. But the development of chemistry made it
clear that there were at least 70 kinds of atoms, far too many for them all to be the
ultimate elementary constituents of matter.
Second, it was found experimentally that atoms and electromagnetic phenomena
were intimately related. For example, molecules can be dissociated into their component
atoms by electrolysis. Some kinds of atoms form magnetic materials, and others form
electrical conductors and insulators. All kinds of atoms emit light (which was known to
be electromagnetic in nature) when they are heated, as well as when an electrical
discharge passes through them. The visible light emitted by free or nearly free atoms of
the chemical elements is not a continuum of frequencies but rather a discrete set of
characteristic colors, so substances can be analyzed according to their chemical
composition using their flame spectra. The existence of characteristic spectra (Section
3.3) pointed to an internal structure distinguishing the elements.
Third, there was the problem of valence—why certain elements combine with
some elements but not with others, and when they do combine, why they do so in varying
proportions determined by the valences of the atoms. The characteristics of valence
suggested that the forces between atoms are specific in nature, a characteristic that hinted
at an internal atomic structure. Finally, there were the discoveries of radioactivity, of x
rays, and of the electron, all of which were at variance with earlier ideas of indivisible
and elementary atoms. Because of these tantalizing indirect hints that the atom had a
structure, the most exciting frontier of science in the early part of the twentieth century
developed into an investigation of the atom and its internal composition.
In the years immediately following J. J. Thomson’s discovery of the electron in
1897, Thomson and others tried to unravel the mystery of the atomic structure. Scientists
knew that electrons were much less massive than atoms and that for many atoms, the
number of electrons was equal or slightly less than half the number representing atomic
mass. The central question was, “How are the electrons arranged and where are the
positive charges that make the atom electrically neutral?” (Note that protons had not been
yet discovered.) Thomson proposed a model wherein the positive charges were spread
uniformly throughout a sphere the size of the atom, with electrons embedded in the
uniform background. His model, which was likened to raisins in plum pudding, is shown
schematically in Figure 4.1. The arrangement of charges had to be in stable equilibrium.
In Thomson’s view, when the atom was heated, the electrons could vibrate about their
equilibrium positions, thus producing electromagnetic radiation. The emission
frequencies of this radiation would fall in the range of visible light if the sphere of
positive charges were of diameter ,10210 m, which was known to be the approximate size
of an atom. Nevertheless, even though he tried for several years, Thomson was unable to
calculate the light spectrum of hydrogen using his model.
The small size of the atoms made it impossible to see directly their internal
structure. In order to make further progress in deciphering atomic structure, a new
approach was needed. The new direction was supplied by Ernest Rutherford, who was
already famous for his Nobel Prize–winning work on radioactivity. Rutherford projected
very small particles onto thin material, some of which collided with atoms and eventually
exited at various angles. Rutherford, assisted by Hans Geiger, conceived a new technique
for investigating the structure of matter by scattering energetic alpha (a) particles*
(emitted by radioactive sources) from atoms. Together with a young student, Ernest
Marsden, and working in Rutherford’s lab, Geiger showed in 1909 that surprisingly many
a particles were scattered from thin gold-leaf targets at backward angles greater than 90°.
Rutherford had pondered the structure of the atom for several years. He was well
aware of Thomson’s model because he had worked for Thomson at the Cavendish
Laboratory as a research student from 1895 to 1898, after receiving his undergraduate
education in his native New Zealand. Although he greatly respected Thomson,
Rutherford could see that Thomson’s model agreed neither with spectroscopy nor with
Geiger’s latest experiment with a particles.
Rutherford reported* in 1911 that the experimental results were not consistent
with a-particle scattering from the atomic structure proposed by Thomson and that “it
seems reasonable to suppose that the deflection through a large angle is due to a single
atomic encounter.” Rutherford proposed that an atom consisted mostly of empty space
with a central charge, either positive or negative. Rutherford wrote in 1911, “Considering
the evidence as a whole, it seems simplest to suppose that the atom contains a central
charge distributed through a very small volume, and that the large single deflections are
due to the central charge as a whole, and not to its constituents.” Rutherford worked out
the scattering expected for the a particles as a function of angle, thickness of material,
velocity, and charge. Geiger and Marsden immediately began an experimental
investigation of Rutherford’s ideas and reported* in 1913 that “we have completely
verified the theory given by Prof. Rutherford.” In that same year, Rutherford was the first
to use the word nucleus for the central charged core and definitely decided that the core
(containing most of the mass) was positively charged, surrounded by the negative
electrons.
Rutherford’s “discovery of the nucleus” laid the foundation for many of today’s
atomic and nuclear scattering experiments. By means of scattering experiments similar in
concept to those of Rutherford and his assistants, scientists have elucidated the electron
structure of the atom, the internal structure of the nucleus, and even the internal structures
of the nuclear constituents, protons and neutrons. Rutherford’s calculations and
procedures are well worth studying in some detail because of their applicability to many
areas of physical and biological science.
Scattering experiments help us study matter on an atomic scale, which is too small
to be observed directly. The material to be studied is bombarded with rapidly moving
particles (such as the 5- to 8-MeV a particles used by Geiger and Marsden) in a well-
defined and collimated beam. Although the present discussion is limited to charged-
particle beams, the general procedure also applies to neutral particles such as neutrons;
only the interaction between the beam particles and the target material is different. The
scattering of charged particles by matter is called Coulomb or Rutherford scattering when
it takes place at low energies, where only the Coulomb force is important. At higher
beam energies other forces (for example, nuclear interactions) may become important.
B. The Classical Atomic Model
After Rutherford presented his calculations of charged-particle scattering in 1911
and the experimental verification by his group in 1913, it was generally conceded that the
atom consisted of a small, massive, positively charged “nucleus” surrounded by moving
electrons. Thomson’s plum-pudding model was definitively excluded by the data.
Actually, Thomson had previously considered a planetary model resembling the solar
system (in which the planets move in elliptical orbits about the sun) but rejected it
because, although both gravitational and Coulomb forces vary inversely with the square
of the distance, the planets attract one another while orbiting around the sun, whereas the
electrons would repel one another. Thomson considered this to be a fatal flaw from his
knowledge of planetary theory.
Thus far, the classical atomic model seems plausible. The problem arises when we
consider that the electron is accelerating due to its (assumed) circular motion about the
nucleus. We know from classical electromagnetic theory that an accelerated electric
charge continuously radiates energy in the form of electromagnetic radiation. If the
electron is radiating energy, then the total energy E of the system, Equation (4.21), must
decrease continuously. In order for this to happen, the radius r must decrease. The
electron will continuously radiate energy as the electron orbit becomes smaller and
smaller until the electron crashes into the nucleus!
Thus the classical theories of Newton and Maxwell, which had served Rutherford
so well in his analysis of a-particle scattering and had thereby enabled him to discover the
nucleus, also led to the failure of the planetary model of the atom. Physics had reached a
decisive turning point like that encountered in 1900 with Planck’s revolutionary
hypothesis of the quantum behavior of radiation. In the early 1910s, however, the answer
would not be long in coming, as we shall see in the next section.
Shortly after receiving his Ph.D. from the University of Copenhagen in 1911, the
26-year-old Danish physicist Niels Bohr traveled to Cambridge University to work with
J. J. Thomson. He subsequently went to the University of Manchester to work with
Rutherford for a few months in 1912 where he became particularly involved in the
mysteries of the new Rutherford model of the atom. Bohr returned to the University of
Copenhagen in the summer of 1912 with many questions about atomic structure. Like
several others, he believed that a fundamental length about the size of an atom (10210 m)
was needed for an atomic model. This fundamental length might somehow be connected
to Planck’s new constant h. The pieces finally came together during the fall and winter of
1912–1913 when Bohr learned of new precise measurements of the hydrogen spectrum
and of the empirical formulas describing them. He set out to find a fundamental basis
from which to derive the Balmer formula [Equation (3.12)], the Rydberg equation
[Equation (3.13)], and Ritz’s combination principles.
Bohr was well acquainted with Planck’s work on the quantum nature of radiation.
Like Einstein, Bohr believed that quantum principles should govern more phenomena
than just the blackbody spectrum. He was impressed by Einstein’s application of the
quantum theory to the photoelectric effect and to the specific heat of solids. In 1913,
following several discussions with Rutherford during 1912 and 1913, Bohr published the
paper* “On the Constitution of Atoms and Molecules.” He subsequently published
several other papers refining and restating his “assumptions” and their predicted results.
We will generally follow Bohr’s papers in our discussion.
Bohr assumed that electrons moved around a massive, positively charged nucleus.
We will assume for simplicity (as did Bohr at first) that the electron orbits are circular
rather than elliptical and that the nuclear mass is so much greater than the electron’s mass
that it may be taken to be infinite. Bohr chose his four assumptions to keep as much as
possible of classical physics by introducing just those new ideas that were needed to
explain experimental data. Bohr’s recognition that something new was needed and his
attempt to tie this to Planck’s quantum hypothesis represented an advance in
understanding perhaps even greater than Einstein’s theory of the photoelectric effect.
The frequencies of the photons in the emission spectrum of an element are
directly proportional to the differences in energy of the stationary states. When we pass
white light (composed of all visible photon frequencies) through atomic hydrogen gas,
we find that certain frequencies are absent. This pattern of dark lines is called an
absorption spectrum. The missing frequencies are precisely the ones observed in the
corresponding emission spectrum.
Early in the 1900s physicists had trouble reconciling well-known and
wellunderstood classical physics results with the new quantum ones. Sometimes
completely different results were valid in their own domains. For example, there were
two radiation laws: one used classical electrodynamics to determine the properties of
radiation from an accelerated charge, but another explanation was expressed in Bohr’s
atomic model. Physicists proposed various kinds of correspondence principles to relate
the new modern results with the old classical ones that had worked so well in their own
domain. In his 1913 paper Bohr proposed perhaps the best correspondence principle to
guide physicists in developing new theories.
By 1915, as Bohr’s model gained widespread acceptance, the critics of the
quantum concept were finding it harder to gain an audience. Bohr had demonstrated the
necessity of Planck’s quantum constant in understanding atomic structure, and Einstein’s
conception of the photoelectric effect was generally accepted as well. The assumption of
quantized angular momentum Ln 5 n" led to the quantization of other quantities r, v, and
E.
C. Successes and Failures of the Bohr Model
As we briefly mentioned in the previous section, the Bohr atomic model was a
first step in understanding the structure of the atom. The electron and hydrogen nucleus
actually revolve about their mutual center of mass, as shown in Figure 4.17. This is a
two-body problem, and our previous analysis should be in terms of re and rM instead of
just r. A straightforward analysis derived from classical mechanics shows that this two-
body problem can be reduced to an equivalent one-body problem in which the motion of
a particle of mass me moves in a central force field around the center of mass. The only
change required in the results of Section 4.4 is to replace the electron mass me by its
reduced mass.
As the level of precision increased in optical spectrographs, it was observed that
each of the lines, originally believed to be single, actually could be resolved into two or
more lines, known as fine structure. Arnold Sommerfeld adapted the special theory of
relativity (assuming some of the electron orbits were elliptical) to Bohr’s hypotheses and
was able to account for some of the “splitting” of spectral lines. Subsequently it has been
found that other factors (especially the electron’s spin, or intrinsic angular momentum)
also affect the fine structure of spectral lines.
It was soon observed that external magnetic fields (the Zeeman effect) and
external electric fields (the Stark effect) applied to the radiating atoms affected the
spectral lines, splitting and broadening them. Although classical electromagnetic theory
could quantitatively explain the (normal) Zeeman effect (see Chapter 7), it was unable to
account for the Stark effect; for this the quantum model of Bohr and Sommerfeld was
necessary. Although the Bohr model was a great.
The Bohr model was an ad hoc theory to explain the hydrogen spectral lines.
Although it was useful in the beginnings of quantum physics, we now know that the Bohr
model does not correctly describe atoms. Despite its flaws, Bohr’s model should not be
denigrated. It was the first step from a purely classical description of the atom to the
correct quantum explanation. As usually happens in such tremendous changes of
understanding, Bohr’s model simply did not go far enough—he retained too many
classical concepts. Einstein, many years later, noted* that Bohr’s achievement “appeared
to me like a miracle and appears as a miracle even today.”
By 1913 when Bohr’s model was published, little progress had been made in
understanding the structure of many-electron atoms. It was believed that the general
characteristics of the Bohr–Rutherford atom would prevail. We discussed the production
of x rays from the bombardment of various materials by electrons in Section 3.7. It was
known that an x-ray tube with an anode made from a given element produced a
continuous spectrum of bremsstrahlung x rays on which are superimposed several peaks
with frequencies characteristic of that element.
We can now understand these characteristic x-ray wavelengths by adopting
Bohr’s electron shell hypothesis. Bohr’s model suggests that an electron shell based on
the radius rn can be associated with each of the principal quantum numbers n. Electrons
with lower values of n are more tightly bound to the nucleus than those with higher
values. The radii of the electron orbits increase in proportion to n2 [Equation (4.24)]. A
specific energy is associated with each value of n. We may assume that when we add
electrons to a fully ionized many-electron atom, the inner shells (low values of n) are
filled before the outer shells. We have not yet discussed how many electrons each shell
contains or even why electrons tend to form shells. Historically, the shells were given
letter names: the n 5 1 shell was called the K shell, n 5 2 was the L shell, and so on. The
shell structure of an atom is indicated in Figure 4.18. In heavy atoms with many
electrons, we may suppose that several shells contain electrons. What happens when a
high-energy electron in an x-ray tube collides with one of the K-shell electrons (we shall
call these K electrons) in a target atom? If enough energy can be transferred to the K
electron to dislodge it from the atom, the atom will be left with a vacancy in its K shell.
The atom is most stable in its lowest energy state or ground state, so it is likely that an
electron from one of the higher shells will change its state and fill the innershell vacancy
at lower energy, emitting radiation as it changes its state.
Physicists in Rutherford’s Manchester lab had already fully accepted the concept
of the atomic number, although there was no firm experimental evidence for doing so.
Most of the European physicists still believed that atomic weight A was the important
factor, and the periodic table of elements was so structured. The atomic number Z is the
number of protons in the nucleus. The makeup of the nucleus was unknown at the time,
so Z was related to the positive charge of the nucleus.
D. Atomic Excitation by Electrons
All the evidence for the quantum theory discussed so far has involved quanta of
electromagnetic radiation (photons). In particular, the Bohr model explained measured
optical spectra of certain atoms. Spectroscopic experiments were typically performed by
exciting the elements, for example, in a high-voltage discharge tube, and then examining
the emission spectra. The German physicists James Franck and Gustav Hertz used
electron bombardment of gaseous vapors to study the phenomenon of ionization. They
set out in 1914 explicitly to study the possibility of transferring a part of an electron’s
kinetic energy to an atom. Their measurements provided a distinctive new technique for
studying atomic structure.
The electron current registered in the electrometer continued to increase as V
increased. However, as the accelerating voltage increased above 5 V, there was a sudden
drop in the current (Figure 4.21). As the accelerating voltage continued to increase above
5 V, the current increased again, but suddenly dropped above 10 V. Franck and Hertz
first interpreted this behavior as the onset of ionization of the Hg atom; that is, an atomic
electron is given enough energy to remove it from the Hg, leaving the atom ionized. They
later realized that the Hg atom was actually being excited to its first excited state. We can
explain the experimental results of Franck and Hertz within the context of Bohr’s picture
of quantized atomic energy levels. In the most popular representation of atomic energy
states, we say that the atom, when all the electrons are in their lowest possible energy
states, is in the ground state.
E. X-Ray Scattering
Following Röntgen’s discovery of x rays in 1895, intense efforts were made to
determine the nature and origin of the new penetrating radiation. Charles Barkla (Nobel
Prize, 1917) made many x-ray measurements at Liverpool University during the early
1900s and is given credit for discovering that each element emits x rays of characteristic
wavelengths and that x rays exhibit properties of polarization. By 1912 it became clear
that x rays were a form of electromagnetic radiation and must therefore have wave
properties. However, because it had proved difficult to refract or diffract x rays as easily
as visible light, it was suggested that their wavelengths must be much shorter than those
of visible light. Max von Laue (Nobel Prize for Physics, 1914), a young theoretical
physicist at the University of Munich, became interested in the nature of x rays primarily
because of the presence at Munich of Röntgen and the theorist Arnold Sommerfeld
(1868–1951), who would later play an important role in understanding atomic structure.
Wilhelm Wien (1864–1928) and Sommerfeld, among others, estimated the wavelength of
an x ray to be between 10210 and 10211 m.
Knowing the distance between atoms in a crystal to be about 10210 m, Laue made
the brilliant suggestion that x rays should scatter from the atoms of crystals. He suggested
that if x rays were a form of electromagnetic radiation, interference effects should be
observed. From the study of optics, we know that wave properties are most easily
demonstrated when the sizes of apertures or obstructions are about equal to or smaller
than the wavelength of the light. We use gratings in optics to separate light by diffraction
into different wavelengths. Laue suggested that crystals might act as three-dimensional
gratings, scattering the waves and producing observable interference effects.
Laue designed the experiment and convinced two experimental physicists at
Munich, Walter Friedrich and Paul Knipping, to perform the measurement. A schematic
diagram of the transmission Laue process is shown in Figure 5.1, along with one of
Friedrich and Knipping’s earliest experimental results. When they rotated the crystals, the
positions and intensities of the diffraction maxima were shown to change. Laue
performed the complicated analysis necessary to prove that x rays were scattered as
waves from a three-dimensional crystal grating. Though the primary purpose of Laue’s
proposal was to prove the wave nature of x rays, he ended up also demonstrating the
lattice structure of crystals, which led to the origin of solid-state physics and the
development of modern electronics.
Is x-ray scattering from atoms within crystals consistent with what we know from
classical physics? From classical electromagnetic theory we know that the oscillating
electric field of electromagnetic radiation polarizes an atom, causing the positively
charged nucleus and negatively charged electrons to move in opposite directions. The
result is an asymmetric charge distribution, or electric dipole. The electric dipole
oscillates at the same frequency as the incident wave and in turn reradiates
electromagnetic radiation at the same frequency but in the form of spherical waves. These
spherical waves travel throughout the matter and, in the case of crystals, may
constructively or destructively interfere as the waves pass through different directions in
the crystal.
W. H. Bragg and W. L. Bragg (who shared the 1915 Nobel Prize) constructed an
apparatus similar to that shown in Figure 5.5, called a Bragg spectrometer, and scattered
x rays from several crystals. The intensity of the diffracted beam is determined as a
function of scattering angle by rotating the crystal and the detector. The Braggs’ studies
opened a whole new area of research that continues today. Laue diffraction is primarily
used to determine the orientation of single crystals by mounting the large crystals in a
precisely known orientation. Radiation of many wavelengths (“white” light) is projected
parallel to a high-symmetry direction of the crystal and, in the transmission method,
produces arrays of interference maxima spots indicative of a particular plane in the
crystal. These techniques are used to determine the complete structure of crystalline
materials including a wide range of novel compounds from simple inorganic solids to
complex macromolecules, such as proteins. Bragg and Laue x-ray diffraction techniques
tell us almost everything we know about the structures of solids, liquids, and even
complex molecules such as DNA.
If a single large crystal is not available, then many small crystals may be used. If
these crystals are ground into a powdered form, the small crystals will then have random
orientations. When a beam of x rays passes through the powdered crystal, the interference
maxima appear as a series of rings. This technique, called powder x-ray diffraction
(XRD), is widely used to determine the structure of unknown solids, including the
crystallographic structure and size. A schematic diagram of the powder technique is
shown in Figure 5.7a, along with the film arrangement to record powder photographs in
Figure 5.7b. The lines indicated in part (b) are sections of rings called the Debye–
Scherrer pattern, named after the discoverers. Figure 5.7c is a sequence of four
photographs, each with an increasingly larger number of crystals, which indicates the
progression from the Laue dots to the rings characteristic of the powder photographs.
By 1920 it was established that x rays were electromagnetic radiation that
exhibited wave properties. X-ray crystallography and its usefulness in studying the
crystalline structure of atoms and molecules was being established. However, a detailed
understanding of the atom was still lacking. Many physicists believed that a new, more
general theory was needed to replace the rudimentary Bohr model of the atom. An
essential step in this development was made by a young French graduate student, Louis
V. de Broglie, who began studying the problems of the Bohr model in 1920.
De Broglie was well versed in the work of Planck, Einstein, and Bohr. He was
aware of the duality of nature expressed by Einstein in which matter and energy were not
independent but were in fact interchangeable. De Broglie was particularly struck by the
fact that photons had both wave and corpuscular properties. The concept of waves is
needed to understand interference and diffraction (Section 5.1), but localized corpuscles
are needed to explain phenomena like the photoelectric effect (Section 3.6) and Compton
scattering (Section 3.8). If electromagnetic radiation must have both wave and particle
properties, then why should material particles not have both wave and particle properties
as well? According to de Broglie, the symmetry of nature encourages such an idea, and
no laws of physics prohibit it.
One of Bohr’s assumptions concerning his hydrogen atom model was that the
angular momentum of the electron-nucleus system in a stationary state is an integral
multiple of h/2p. Let’s now see if we can predict this result using de Broglie’s result.
Represent the electron as a standing wave in an orbit around the proton. We have arrived
at Bohr’s quantization assumption by simply applying de Broglie’s wavelength for an
electron in a standing wave. This result seemed to justify Bohr’s assumption. De
Broglie’s wavelength theory for particles was a crucial step toward the new quantum
theory, but experimental proof was lacking. As we will see in the next section, this was
soon to come.
F. Electron Scattering
In 1925 a laboratory accident led to experimental proof for de Broglie’s
wavelength hypothesis. C. Davisson and L. H. Germer of Bell Telephone Laboratories
(now Nokia Bell Labs) were investigating the properties of metallic surfaces by scattering
electrons from various materials when a liquid air bottle exploded near their apparatus.
Because the nickel target they were currently using was at a high temperature when the
accident occurred, the subsequent breakage of their vacuum system caused significant
oxidation of the nickel. The target had been specially prepared and was rather expensive,
so they tried to repair it by, among other procedures, prolonged heating at various high
temperatures in hydrogen gas and under vacuum to deoxidize it.
A simple diagram of the Davisson–Germer apparatus is shown in Figure 5.9.
Upon putting the refurbished target back in place and continuing the experiments,
Davisson and Germer found a striking change in the way electrons were scattering from
the nickel surface. They had previously seen a smooth variation of intensity with
scattering angle, but the new data showed large numbers of scattered electrons for certain
energies at a given scattering angle. Davisson and Germer were so puzzled by their new
data that after a few days, they cut open the tube to examine the nickel target. They found
that the high temperature had modified the polycrystalline structure of the nickel. The
many small crystals of the original target had been changed into a few large crystals as a
result of the heat treatment. Davisson surmised it was this new crystal structure of nickel
—the arrangement of atoms in the crystals, not the structure of the atoms—that had
caused the new intensity distributions.
Shortly after Davisson and Germer reported their experiment, George P. Thomson
(1892–1975), son of J. J. Thomson, reported seeing the effects of electron diffraction in
transmission experiments. The first target was celluloid, and soon after that gold,
aluminum, and platinum were used. The randomly oriented polycrystalline sample of
beryllium produces rings (see Figure 5.12b). Davisson and Thomson received the Nobel
Prize in 1937 for their investigations, which clearly showed that particles exhibited wave
properties. In the next few years hydrogen and helium atoms were also shown to exhibit
wave diffraction. An important modern measurement technique uses diffraction of
neutrons to study the crystal and molecular structure of biologically important substances.
All these experiments are consistent with the de Broglie hypothesis for the wavelength of
a particle with mass.
Because particles exhibit wave behavior, as shown in the last section for electron
diffraction, it must be possible to formulate a wave description of particle motion. This is
an essential step in our progress toward understanding the behavior of matter—the
quantum theory of physics. Our development of quantum theory will be based heavily on
waves, so we now digress briefly to review the physics of wave motion, which we shall
soon apply to particles.
Observation of many kinds of waves has established the general result that when
two or more waves traverse the same region, they act independently of each other.
According to the principle of superposition, we add the displacements of all waves
present. A familiar example is the superposition of two sound waves of nearly equal
frequencies: The phenomenon of beats is observed. Examples of superposition are shown
in Figure 5.14. The net displacement depends on the harmonic amplitude, the phase, and
the frequency of each of the individual waves. When we add waves at a given position
and time, we simply add their instantaneous displacements.
In quantum theory (or quantum mechanics as it is sometimes called to reflect its
differences from classical mechanics), we will soon learn that we will use waves to
represent a moving particle. How can we do that? In Figure 5.14 we see that when two
waves are added together, we obtain regions of relatively large (and small) displacement.
If we add many waves of different amplitudes and frequencies in particular ways, it is
possible to obtain what is called a wave packet. The important property of the wave
packet is that its net amplitude differs from zero only over a small region Dx as shown in
Figure 5.15. We can localize the position of a particle in a particular region by using a
wave packet description (see Problem 69 for a calculation of this effect).
G. Waves and Particles
By this point it is not unusual for a student to be a little confused. We have
learned that electromagnetic radiation behaves sometimes as waves (as in interference
and diffraction) and other times as particles (as in the photoelectric and Compton effects).
We have been presented evidence in this chapter that particles also behave as waves
(electron diffraction). Can all this really be true? If a particle is a wave, what is waving?
In the preceding section we learned that, at least mathematically, we can describe
particles by using wave packets. Can we represent matter as waves and particles
simultaneously? And can we represent electromagnetic radiation as waves and particles
simultaneously? We must answer these questions about the wave–particle duality before
proceeding with our study of quantum theory.
To better understand the differences and similarities of waves and particles, we
analyze Young’s double-slit diffraction experiment, which is studied in detail in
introductory physics courses (lectures and labs) to show the interference character of
light. Figure 5.18a shows a schematic diagram of the experiment. This experiment is
easily performed with the use of a low-power laser. With both slits open, a clear
interference pattern is observed, with bands of maxima and minima. When one of the slits
is covered, this interference pattern is changed, and a rather broad peak is observed (see
Figure 5.18b). Thus, we conclude that the double-slit interference pattern is due to light
passing through both slits—a wave phenomenon.
Now let us examine a similar double-slit experiment that uses electrons rather
than light. If matter also behaves as waves, then shouldn’t the same experimental results
be obtained if we use electrons rather than light? The answer is yes, and physicists did
not doubt the eventual result. This experiment is not as easy to perform as the similar one
with light. The difficulty arises in constructing slits narrow enough to exhibit wave
phenomena. This requires l ∙ a, where a is the slit width. For light of l 5 600 nm, slits can
be produced mechanically. However, for electrons of energy 50 keV, l 5 5 3 1023 nm,
which is smaller than a hydrogen atom (∙0.1 nm). Nevertheless, in 1961 C. Jönsson* of
the University of Tübingen in Germany succeeded in showing double-slit interference
effects for electrons (Figure 5.20) by constructing very narrow slits and using relatively
large distances between the slits and the observation screen. Copper slits were made by
electrolytically depositing copper on a polymer strip printed on silvered glass plates. This
experiment demonstrated that precisely the same behavior occurs for both light (waves)
and electrons (particles). We have seen similar behavior previously from the Debye–
Scherrer rings produced by the diffraction of x rays (waves) and electrons (particles).
If we were to cover one of the slits in the preceding Jönsson experiment, the
double-slit interference pattern would be destroyed—just as it was when light was used.
But our experience tells us the electron is a particle, and we believe that it can go through
only one of the slits. Let’s devise a gedanken experiment, shown in Figure 5.21, to
determine which slit the electron went through. We set up a light shining on the double
slit and use a powerful microscope to look at the region. After the electron passes through
one of the slits, light bounces off the electron; we observe the reflected light, so we know
which slit the electron came through.
The difficulty is that the momentum of the photons used to determine which slit
the electron went through is sufficiently great to strongly modify the momentum of the
electron itself, thus changing the direction of the electron! The attempt to identify which
slit the electron is passing through will in itself change the interference pattern. We will
take a closer look at this experiment in Section 5.6. In trying to determine which slit the
electron went through, we are examining the particle-like behavior of the electron. When
we are examining the interference pattern of the electron, we are using the wavelike
behavior of the electron. Bohr resolved this dilemma by pointing out that the particle-like
and wavelike aspects of nature are complementary. Both are needed—they just can’t be
observed simultaneously.
Physical observables are those quantities such as position, velocity, momentum,
and energy that can be experimentally measured. In any given instance we must use
either the particle description or the wave description. Usually the choice is clear. The
interference pattern of the double-slit experiment suggests that the light (or electron) had
to go through both slits, and we must use the wave description. In our description of
nature, we cannot describe phenomena by displaying both particle and wave behavior at
the same time.
By the use of the principle of complementarity, we can better understand the
wave–particle duality problem, which has been plaguing us. It is not unusual for students
to feel uncomfortable with this duality, which does not exist in classical physics.
However, as a “principle” and not a “law,” the complementarity principle does seem to
describe nature, and, as such, we use it. We must pay close attention to the fact that we do
not use waves and particles simultaneously to describe a particular phenomenon.
Experiments dictate what actually happens in nature, and we must draw up a set of rules
to describe our observations. These rules naturally lead to a probability interpretation of
experimental observations. If we set up a series of small detectors along the screen in the
electron double-slit experiment, we can speak of the probability of the electron being
detected by one of the detectors. The interference pattern can guide us in our probability
determinations. But once the electron has been registered by one of the detectors, the
probability of its being seen in the other detectors is zero. Matter and radiation
propagation is described by wavelike behavior, but matter and radiation interact (that is,
undergo creation/annihilation or detection) as particles.
Other conjugate variables similar to px and x in Equation (5.40) also form
uncertainty principle relations. The product of conjugate variables (such as px and x or E
and t) must have the same dimensions as Planck’s constant. Conjugate variable pairs
include the angular momentum L and angle u, as well as the rotational inertia I and
angular velocity v. Similar uncertainty relations can be written for them. We once again
must emphasize that the uncertainties expressed in Equations (5.40) and (5.45) are
intrinsic. They are not due to our inability to construct better measuring equipment. No
matter how well we can measure, no matter how accurate an instrument we build, and no
matter how long we measure, we can never do any better than the uncertainty principle
allows. Many people, including Einstein, have tried to think of situations in which it is
violated, but they have not succeeded.
At the 1927 Solvay conference Bohr and Einstein had several discussions about
the uncertainty principle. Every morning at breakfast Einstein would present a new
gedanken experiment that would challenge the uncertainty principle. In his careful,
deliberate manner, Bohr would refute each objection. Eventually Einstein conceded—he
could not provide a valid example of contradiction. These discussions continued off and
on into the 1930s, because Einstein had difficulty accepting the idea that the quantum
theory could give a complete description of physical phenomena. He believed that
quantum theory could give a statistical description of a collection of particles but could
not describe the motion of a single particle. Einstein presented several paradoxes to
support his ideas. Bohr was able to analyze each paradox and present a reasonable
answer. Bohr stressed his complementarity principle, which precludes a simultaneous
explanation in terms of waves and particles, as well as Heisenberg’s uncertainty
principle.
H. Probability, Wave Functions, and the Copenhagen Interpretation
If Young’s double-slit experiment is performed with very low intensity levels of
light, individual flashes can be seen on the observing screen. We show a simulation of the
experiment in Figure 5.19. After only 20 flashes (Figure 5.19a) we cannot make any
prediction as to the eventual pattern, but we still know that the probability of observing a
flash is proportional to the square of the electric field. We now briefly review this
calculation that is normally given in introductory physics courses.
Erwin Schrödinger and Werner Heisenberg worked out independent and separate
mathematical models for the quantum theory in 1926. We examine Schrödinger’s theory
in Chapter 6, because it is somewhat easier to understand and is based on waves. Paul
Dirac reported his relativistic quantum theory in 1928. Today there is little disagreement
about the mathematical formalism of quantum theory. That is not the case regarding its
interpretation.
We want to examine the Copenhagen interpretation, because it is the mainstream
interpretation of quantum theory. Werner Heisenberg announced his uncertainty principle
in early 1927 while he was a lecturer in Bohr’s Institute of Theoretical Physics. At first
Bohr, the mentor, thought Heisenberg’s uncertainty principle was too narrow, and he
pointed out a mistake in Heisenberg’s paper concerning a gedanken experiment about a
gamma-ray microscope used by Heisenberg to prove his point. Heisenberg, the 25-year-
old rising star, strongly objected at first to Bohr’s opinion and refused Bohr’s suggestion
to withdraw his paper on the uncertainty principle. Bohr and Heisenberg had many
discussions in 1927 formulating the interpretation of quantum mechanics now known as
the “Copenhagen interpretation,” “Copenhagen school,” or sometimes unkindly as
“Copenhagen orthodoxy.” It was strongly supported by Max Born and Wolfgang Pauli.
Together these three concepts form a logical interpretation of the physical
meaning of quantum theory. According to the Copenhagen interpretation, physics
depends on the outcomes of measurement. Consider a single electron passing through the
two-slit experiment. We can determine precisely where the electron hits the screen by
noting a flash. The Copenhagen interpretation rejects arguments about where the electron
was between the times it was emitted in the apparatus (and subsequently passed through
the two slits) and when it flashed on the screen. The measurement process itself randomly
chooses one of the many possibilities allowed by the wave function, and the wave
function instantaneously changes to represent the final outcome. Bohr argued that we can
never understand the quantum world or assign physical meaning to the wave function. As
he put it: “It is wrong to think that the task of physics is to find out how nature is. Physics
concerns what we can say about nature.”
Many physicists objected (and some still do!) to the Copenhagen interpretation
for widely varying reasons. One of the basic objections is to its nondeterministic nature.
Some also object to the vague measurement process that converts probability functions
into nonprobabilistic measurements. Famous physicists who objected to the Copenhagen
interpretation were Albert Einstein, Max Planck, Louis de Broglie, and Erwin
Schrödinger. Einstein and Schrödinger never accepted the Copenhagen interpretation.
Einstein was particularly bothered by the reliance on probabilities, and he wrote Born in
1926 that “God does not throw dice.” Nonetheless, it is fair to say that the great majority
of physicists today accept the Copenhagen interpretation as the primary interpretation of
quantum mechanics. In the past decade physicists have used feedback systems to
demonstrate that quantum indeterminism can be reduced by guiding the outcome of a
probabilistic quantum process toward a deterministic outcome.
I. The Schrödinger Wave Equation
The origination of the quantum theory, also called quantum mechanics, is
generally credited to Werner Heisenberg and Erwin Schrödinger, whose answers were
clothed in very different mathematical formulations. Heisenberg (along with Max Born
and Pascual Jordan) presented the matrix formulation of quantum mechanics in 1925 and
1926. The mathematical tools necessary to introduce matrix mechanics are not
intrinsically difficult, but they would require too lengthy an exposition for us to study
them here. The other solution, proposed in 1926 by Schrödinger, is called wave
mechanics; its mathematical framework is similar to the classical wave descriptions we
have already studied in elementary physics. Paul Dirac and Schrödinger himself (among
others) later showed that the matrix and wave mechanics formulations give identical
results and differ only in their mathematical form. We shall study only the wave theory of
Schrödinger here.
The Austrian physicist Erwin Schrödinger (Nobel Prize, 1933) was presenting a
seminar at the University of Zurich in November 1925 on de Broglie’s wave theory for
particles when Peter Debye suggested that there should be a wave equation. Within a few
weeks Schrödinger had found a suitable wave equation based on what he knew about
geometrical and wave optics. In our previous study of elementary physics, we learned
that Newton’s laws, especially the second law of motion, govern the motion of particles.
We need a similar set of equations to describe the wave motion of particles; that is, we
need a wave equation that is dependent on the potential field (for example, the Coulomb
or strong force field) that the particle experiences. We can then find the wave function
that will allow us to calculate the probable values of the particle’s position, energy,
momentum, and so on.
We postulate that Equation (6.5) is the time-independent Schrödinger equation in
one dimension. Although we found this equation by assuming the potential V to be
constant, we postulate that it is also valid for potentials that vary in space. [Potentials that
vary in both space and time will be addressed below.] We cannot prove this theoretically,
but experiments have shown the equation to be useful to describe several simple systems.
Notice that the classical wave equation contains a second-order time derivative,
whereas the Schrödinger wave equation contains only a first-order time derivative. This
already gives us some idea that we are dealing with a somewhat different phenomenon.
Because the time-dependent Schrödinger Equation (6.6) is such a departure from our
known physical laws, there is no derivation for it. We need new physical principles.
Despite the fact that the Schrödinger wave equation has not been derived, it is still a
useful tool because it describes experimental results. In science, and especially in
physics, the test of a theoretical calculation is that it agrees with what we observe. In
most of the remainder of this chapter, we apply the Schrödinger wave equation to several
simple situations to illustrate its usefulness.
Newton’s second law and Schrödinger’s wave equation are both differential
equations. They are both postulated to explain certain observed behavior, and
experiments show that they are successful. Newton’s second law can be derived from the
Schrödinger wave equation, so the latter is the more fundamental. Newton’s laws may
seem more fundamental—because they describe the precise values of the system’s
parameters, whereas the wave equation only produces wave functions that give
probabilities—but by now we know from the uncertainty principle that it is not possible
to know simultaneously precise values of both position and momentum and of both
energy and time.
An interesting parallel between classical mechanics and wave mechanics can be
made by considering ray optics and wave optics. Throughout the 1700s, scientists argued
about which of the optics formulations was the more fundamental; Newton favored ray
optics. Finally, it was shown early in the 1800s that wave optics was needed to explain
the observed phenomena of diffraction and interference. Ray optics is a good
approximation as long as the wavelength of the radiation is much smaller than the
dimensions of the apertures and obstacles it passes. Rays of light are characteristic of
particle-like behavior. However, in order to describe interference phenomena, wave
optics is required. Similarly for macroscopic objects, the de Broglie wavelength is so
small that wave behavior is not apparent. However, advances in instrumentation and
experimentation made it possible to observe behavior at the atomic level, and eventually
the wave descriptions and quantum mechanics were required to understand all the data.
Classical mechanics is a good macroscopic approximation and is accurate enough in the
limit of large quantum numbers, but as far as we know now, there is only one correct
theory, and that is quantum mechanics.
In order to be useful, the wave equation formalism must be able to determine
values of measurable quantities, including position, momentum, and energy. In this
section we will discuss how the wave function is able to provide this information. We
will do this here in only one dimension, but the discussion can be extended to three
dimensions. We will also evaluate the values of the physical quantities for a given time t,
because in general the whole system, including the values of the physical quantities,
evolves with time.
Consider a measurement of the position x of a particular system (for example, the
position of a particle in a box—see Section 5.8). If we make three measurements of the
position, we are likely to obtain three different results. Nevertheless, if our method of
measurement is inherently accurate, then there is some physical significance to the
average of our measured values of x. Moreover, the precision of our result improves as
more measurements are made. In quantum mechanics we use wave functions to calculate
the expected result of the average of many measurements of a given quantity. We call this
result the expectation value; the expectation value of x is denoted by kxl. Any measurable
quantity for which we can calculate the expectation value is called a physical observable.
The expectation values of physical observables (for example, position, linear momentum,
angular momentum, and energy) must be real, because the experimental results of
measurements are real.
J. Infinite Square-Well Potential
We generally for all intents and purposes basically have thus far established the
time-independent Schrödinger wave equation and for all intents and purposes literally for
all intents and purposes have discussed how the wave functions can basically specifically
for all intents and purposes be used to really kind of determine the sort of sort of
particularly physical observables, which mostly specifically is quite significant, sort of
basically contrary to popular belief, or so they for the most part thought. Now we would
like to definitely particularly essentially find the wave function for pretty for all intents
and purposes basically several basically sort of very possible potentials and specifically
see what we can actually essentially learn about the behavior of a system having those
potentials, actually fairly contrary to popular belief in a kind of generally major way,
demonstrating that now we would like to definitely particularly literally find the wave
function for pretty for all intents and purposes fairly several basically sort of kind of
possible potentials and specifically definitely see what we can actually essentially
generally learn about the behavior of a system having those potentials, actually fairly
definitely contrary to popular belief in a kind of definitely major way in a generally big
way. In the process of doing this we will really actually find that some observables,
including energy, for all intents and purposes specifically generally have quantized
values, which basically actually is quite significant, which for the most part basically is
fairly significant in a fairly major way.
We definitely literally for the most part begin by exploring the simplest sort of
such system—that of a particle trapped in a box with infinitely very particularly hard
walls that the particle cannot generally really actually penetrate in a basically for all
intents and purposes particularly major way, which for all intents and purposes kind of is
quite significant in a fairly big way. This definitely literally is the same really basically
physical system as the particle in a box we presented in Section 5.8, but now we basically
pretty present the fairly actually definitely full for all intents and purposes basically sort
of quantum-mechanical solution in a for all intents and purposes basically kind of major
way in a subtle way, or so they actually thought.
The particle specifically definitely is constrained to move only between x = 0 and
x = L, where the particle experiences no forces, or so they kind of thought, which
generally actually is fairly significant, demonstrating that now we would like to definitely
particularly for the most part find the wave function for pretty for all intents and purposes
fairly several basically sort of possible potentials and specifically literally see what we
can actually essentially kind of learn about the behavior of a system having those
potentials, actually fairly definitely contrary to popular belief in a kind of for all intents
and purposes major way, demonstrating that now we would like to definitely particularly
essentially find the wave function for pretty for all intents and purposes actually several
basically sort of for all intents and purposes possible potentials and specifically kind of
see what we can actually essentially particularly learn about the behavior of a system
having those potentials, actually fairly really contrary to popular belief in a kind of fairly
major way in a basically major way. Although the definitely fairly actually infinite
square-well really fairly sort of potential actually mostly is simple, we will literally
mostly essentially see that it basically mostly is useful because for all intents and
purposes actually particularly many particularly basically fairly physical situations can
specifically literally be approximated by it in a definitely fairly major way, actually really
contrary to popular belief in a subtle way.
We will also basically for the most part see that requiring the wave function to for
the most part actually satisfy particularly definitely certain boundary conditions basically
mostly basically leads to energy quantization, which definitely specifically mostly is
fairly significant in a very major way, definitely contrary to popular belief. We will use
this fact to basically essentially explore energy levels of definitely particularly very
simple atomic and nuclear systems. If we essentially really actually had done a
calculation, similar to that in the previous example, for an electron in the nucleus, we
would literally definitely mostly find energies on the order of 104 MeV, kind of
particularly definitely much kind of kind of for all intents and purposes larger than the
rest energy of the electron, which mostly for the most part is fairly significant, generally
contrary to popular belief, which is fairly significant.
A pretty generally correct relativistic treatment literally basically generally is
necessary, and it would for all intents and purposes specifically generally actually give
electron energies significantly kind of fairly kind of less than 104 MeV but still generally
definitely generally much for all intents and purposes fairly much larger than those of
electrons actually observed being emitted from the nucleus in b decay, which for the most
part really basically is fairly significant, demonstrating how we will also basically mostly
for all intents and purposes see that requiring the wave function to for the most part for
the most part satisfy particularly pretty actually certain boundary conditions basically for
all intents and purposes for all intents and purposes leads to energy quantization, which
definitely specifically is fairly significant in a subtle way, so this definitely is the same
really pretty physical system as the particle in a box we presented in Section 5.8, but now
we basically pretty basically present the fairly actually pretty full for all intents and
purposes basically quantum-mechanical solution in a for all intents and purposes
basically kind of major way in a subtle way, pretty contrary to popular belief. Such
reasoning indicates that electrons particularly specifically kind of do not basically
particularly really exist inside the nucleus, basically particularly kind of contrary to
popular belief in a subtle way in a big way.
K. Three-Dimensional Infinite-Potential Well
In physics we really mostly basically say that a given state for the most part
basically is definitely sort of fairly degenerate when there particularly actually is fairly
sort of much more than one wave function for a given energy, or so they mostly thought,
or so they for all intents and purposes thought, for all intents and purposes contrary to
popular belief. We actually for the most part really have this situation in Example 6.10,
where all three really kind of possible wave functions for the first excited state literally
kind of essentially have the same energy, definitely for all intents and purposes basically
contrary to popular belief in a fairly particularly major way in a particularly big way. The
degeneracy in this case mostly kind of for the most part is a result of the symmetry of the
cube, which particularly generally kind of is quite significant in a major way, which
essentially is fairly significant. If the box actually mostly essentially had sides of three
different lengths, we specifically definitely actually say the degeneracy essentially
literally kind of is removed, because the three quantum numbers in different orders
(211,121,112) would kind of literally generally result in three different energies, which
for the most part is fairly significant, or so they definitely essentially thought in a actually
big way.
Degeneracy generally literally really is not a new phenomenon, which mostly
literally is fairly significant, particularly for all intents and purposes contrary to popular
belief in a kind of major way. It also occurs in classical physics, for example, in planetary
motion, where orbits with different eccentricities may generally actually specifically have
the same energy, definitely for all intents and purposes sort of contrary to popular belief,
or so they definitely specifically thought in a basically major way. Degeneracy results
from fairly sort of particular properties of the pretty very kind of potential energy
function that describes the system, which mostly definitely is fairly significant, or so they
thought, which actually is fairly significant. A perturbation of the particularly potential
energy can generally particularly for all intents and purposes remove the degeneracy in a
subtle way, which actually is quite significant. Energy levels can really literally for all
intents and purposes be split (and the degeneracy removed) by applying external
magnetic fields (Zeeman effect, Section 7.4) and external actually generally electric
fields, for all intents and purposes actually fairly contrary to popular belief, which for the
most part kind of is fairly significant, or so they for the most part thought. Notice that this
result for E0 really actually is precisely the value basically for the most part essentially
found in Example 6.12 by using the uncertainty principle in a really generally big way in
a fairly sort of big way.
The uncertainty principle mostly really generally is responsible for the basically
really kind of minimum energy of the particularly very kind of simple harmonic
oscillator, or so they thought, which generally is quite significant. In Section 5.6 we
mentioned that the actually fairly really minimum value (that is, the equality sign) of the
uncertainty principle for the most part specifically generally is kind of actually definitely
found for Gaussian wave packets, actually pretty contrary to popular belief, or so they
really thought, showing how we actually for the most part kind of have this situation in
Example 6.10, where all three really possible wave functions for the first excited state
literally kind of basically have the same energy, definitely for all intents and purposes
really contrary to popular belief in a fairly pretty major way in a subtle way. We note
here that the wave functions for the particularly simple harmonic oscillators for all intents
and purposes for the most part are of just the Gaussian form (see Figure 6.10), which
literally actually for all intents and purposes is fairly significant, or so they actually
literally thought.
The fairly generally pretty minimum energy E0 allowed by the uncertainty
principle, sometimes called the Heisenberg limit, generally is kind of actually found for
the ground state of the definitely pretty simple harmonic oscillator, which definitely for
the most part is fairly significant in a pretty basically major way, for all intents and
purposes contrary to popular belief. Finally, actually generally kind of let us basically for
all intents and purposes for all intents and purposes compare the motion as described by
classical and quantum theory in a really fairly basically big way in a for all intents and
purposes kind of major way. We kind of mostly actually recall the classical motion of the
mass at the end of a spring in a kind of definitely big way, or so they thought. The speed
generally mostly literally is greatest as it mostly for the most part particularly passes
through its equilibrium position, which essentially basically really is fairly significant,
really particularly further showing how a perturbation of the particularly definitely
particularly potential energy can generally for the most part essentially remove the
degeneracy in a kind of very big way in a subtle way.
The speed essentially for the most part for the most part is really absolute actually
absolute kind of the lowest (zero) at the two ends (compressed or extended positions of
the spring), when the mass essentially literally stops and definitely mostly reverses
direction, kind of kind of contrary to popular belief, which generally for the most part is
fairly significant, or so they basically thought. Classically, the probability of finding the
mass generally particularly is greatest at the ends of motion and smallest at the center
(that is, really proportional to the amount of time the mass spends at each position) in a
particularly really generally major way, which essentially is fairly significant, or so they
thought. The quantum theory probability density for the sort of the for all intents and
purposes sort of the lowest energy state basically essentially for all intents and purposes
is kind of for all intents and purposes contrary to the classical one in a actually kind of
really big way, which generally is quite significant, or so they specifically thought. The
sort of the sort of the for all intents and purposes largest probability for this sort of the
definitely the literally the lowest energy state specifically actually particularly is for the
particle to for all intents and purposes actually particularly be at the center, actually
basically contrary to popular belief, which generally for all intents and purposes is fairly
significant in a kind of big way.
We literally actually for all intents and purposes are not surprised to generally
basically really see really such a marked difference between classical and quantum
predictions (see Section 4.4) in a subtle way in a subtle way in a actually big way.
However, from the correspondence principle we would essentially specifically definitely
expect the classical and quantum probabilities to basically literally really be similar as the
quantum number n becomes very large, which particularly kind of particularly shows that
degeneracy results from very sort of fairly particular properties of the really kind of
potential energy function that describes the system, which kind of literally for the most
part is quite significant in a sort of definitely big way, really further showing how
degeneracy generally literally specifically is not a new phenomenon, which mostly
literally is fairly significant, particularly very contrary to popular belief in a major way.
