Download Unit 8 Waves: Quantum Mechanical Waves

PHYS 302: Unit 8 Quantum Mechanics Supplement – Athabasca University
Unit 8 Waves: Quantum Mechanical Waves
After completing this Unit 8 supplement, you should be able to
1.
identify that particles have a wavelength, and define the de Broglie wavelength.
2.
identify that wave effects similar to those explored in this course also apply to particles on some scale.
3.
identify that free particles can have any energy, momentum, and wavelength.
Introduction
Historically, light had been regarded as a particle stream, a view established in the late 1600s by Newton and
dramatically challenged by Young’s experiments in the early 1800s. Young’s experiments demonstrated that light
obeys the principle of superposition. If light is made up of particles, it is easy to imagine more particles resulting in
more light. However, with waves, the crest of one wave can line up with the trough of another, so that two light
beams can combine, under the right circumstances, to give no light. This phenomenon, known as destructive
interference, will be discussed quantitatively near the end of the course. Since Young observed destructive
interference of light—hard to imagine in a particle-based theory of light, though destructive interference is an
intrinsic property of waves—the prevailing view in the 1800s was that light was, in fact, a wave. This view received
further credence when Maxwell, in the late1800s, showed that electricity and magnetism are not separate forces and
that their union (electromagnetism) completely explained the (then) known properties of light and other forms of
electromagnetic radiation.
Quantization of Energy in Electromagnetic Radiation
By the early twentieth century, the details of Maxwell’s theory had begun to fray. A complex argument allowed
Planck to explain emission of radiation only if it was quantized (i.e., came in small bursts). Atomic theory, based
largely on chemical experiments, had led to the realization that matter came in small chunks, now known as atoms.
By the late 1800s, experiments with “vacuum” tubes had led to the conclusion that electricity flowed in thin gasses
through the action of “corpuscular rays” that travelled in straight lines. In 1906, J. J. Thomson received the Nobel
Prize in Physics for his 1897 conclusion that corpuscular rays were made up of small, negatively-charged particles
that we now call electrons. The emission of electrons from the surfaces of certain materials when illuminated—the
photoelectric effect—implied a link between the particulate properties of electrons and that of light. Photoelectric
emission causes an electric current to flow out of a surface. As expected, such a current increased with an increasing
level of illumination, but light shorter than a certain wavelength (the particular wavelength varied with the material)
was required to cause any emission of electrons, no matter how bright the illumination. This was unexpected. This
evidence strongly suggested that one particle of light ejected one electron, but only if that particle of light had
enough energy to bump the electron out of the surface. This realization—not the special theory of relativity—also
published in 1905, earned Einstein the 1921 Nobel Prize in Physics. The particle of light is called the photon, and
the existence of photons means that light is quantized. Each photon carries an energy E=hν, where h is Planck’s
constant, 6.626068 × 10-34 J s, and ν is the light frequency in Hz. The units of h are joule-seconds, so that when
multiplied by the frequency in Hz, energy in joules results. As you know, light has a wavelength of roughly 500 nm,
and it moves at the speed c=3×108 m/s. Thus, one second worth of light contains roughly 3×108/5×10-7, or 1015,
wavelengths. The number of wavelengths that go by in a second is the frequency, so the frequency of light is very
roughly 1015 Hz. Using E=hν, this means that each photon has, roughly, an energy of 10-20 J, which is very small by
macroscopic standards. In bright sunlight, there is about 1000 W/m2 of energy flux in the visible wavelengths. Since
a watt is a joule per second, this means that 1023 photons pass through a square metre each second on a sunny day.
1 PHYS 302: Unit 8 Quantum Mechanics Supplement – Athabasca University
This is comparable to the number of atoms in a mole—a macroscopic amount of material. Much like a given
material seems continuous despite being made of very small atoms, light also appears continuous in many ways, so
the quantization is not obvious.
Quantization of Momentum in Electromagnetic Radiation
In formulating electromagnetic theory in 1871, Maxwell immediately predicted that light carried momentum. The
resulting radiation pressure was not correctly demonstrated until 1901. (Note: The “light mill” or Crookes
radiometer sometimes said to show radiation pressure is actually made to spin by interaction with gas.) The wave
theory of light correctly predicts the size of radiation pressure using Maxwell’s equations. Knowing that light also
has particle properties allows for a more direct understanding of radiation pressure, since the fact that moving
particles have momentum is one of the first topics studied in an introductory physics course. In the case of photons,
the momentum p=hk/2π, where k is the wave number. Since the factor 2π appears frequently, we often use the
symbol ħ (h-bar), the reduced Planck’s constant, which is h/2π, and has a value very close to 10-34 J s. We should
also write the momentum and wave number as vectors, indicated by bold type. Thus, p=ħk. We can write the energy
relation in a form echoing this as E=ħω. Recall that our preferred symmetric form for travelling waves is
y(x,t)=Asin(kx-ωt) (if the initial phase is 0). At points in the wave having the same phase kx-ωt, the wave value is
the same. Thus, if kx-ωt=b, with b constant, whatever part of the wave which that phase value corresponds to moves
along, and x 

k
b  t
k
 bk  k t . By analogy to a particle in motion at constant speed, where x=x0+vt, we can identify
as the wave speed associated with change of phase, or the phase speed. For light, the phase speed is the speed of
light, c= 2.99792458×108 m/s. So, c 
to write this is p 
h
c

k
, that is, k 

c
. In scalar form, we can write p 
ħ
c
 Ec . Another way
 h . The higher the frequency of a wave, the more energetic its photons, and the more
momentum they carry. As such, short wavelengths correspond to high energy and momentum. In many ways,
X-rays, for example, are fairly particle-like: each photon carries a lot of momentum, and the wavelength is so small
that wave effects are not evident. Photons of this energy or greater behave like particles. However, since photons
carry no charge, X-rays do not respond to electric or magnetic fields.
de Broglie Waves
It is natural to think of matter as being quantized, since the idea of atoms and particles such as electrons is familiar.
As with light, however, the fine subdivision of matter is not totally obvious, and we often deal with bulk matter in
solid, fluid, or gas form. Louis de Broglie (pronounced roughly “de Broy”) hypothesized in his doctoral thesis that
material particles have wave properties. He won a Nobel prize for this in 1929. In particular, de Broglie indicated
that the relations E=ħω and p=ħk should apply to material particles just as to photons, implying that a wave vector
and a frequency are associated with particles. We can take the analogy with light further if we generalize p 
Then we can calculate a wavelength associated with a particle: the de Broglie wavelength

h
p
h

.
. For low-energy
particles (slow-moving compared to c), the momentum (from classical mechanics) is approximately p=mv, where m
is the particle mass and v is its speed. In this case, we can write
  mvh . Lighter particles correspond to a larger de
Broglie wavelength, and electrons are the lightest commonly measured particle. Electrons may be accelerated (gain
speed) by being sent through an electric field, and their energy change is proportional to the change in electric
potential between their starting and finishing locations. Thus, it is convenient to measure electron energies in
“electron volts” or eV. An old-style tube television may use electrons having gone through up to 10,000 V. A
famous laboratory experiment used much slower, less energetic electrons. From the formula, you can see that
slower, more sluggish electrons will give a longer wavelength. In the experiment, 50 eV electrons had a wavelength
2 PHYS 302: Unit 8 Quantum Mechanics Supplement – Athabasca University
of only about 10-10 m, about the same as the spacing between atoms in a solid, clearly showing that the wavelength
of particles is important only on very small scales. While this is interesting, is it speculation or fact?
Experimentation showed the basic factual nature of associating wave properties with electrons and ultimately with
other particles. De Broglie published his thesis in 1924, but evidence of the wave nature of electrons was not found
until early 1927.
Although we have not yet studied diffraction in this course, we can imagine that waves scattered from various points
may meet at a distant point with their crests lined up with other wave crests, or instead, with other wave troughs. If
the crests line up, the wave amplitude increases; if the crests line up with the troughs, the waves can add to give
nothing (the destructive interference mentioned above). If electrons were particles, there might be more particles in
one direction than in another after scattering, but complete cancellation would not be possible. Davisson and
Germer’s experiments on reflection of electrons from crystals and G. P. Thomson’s experiments on transmission of
electrons led to Davisson and Thomson sharing the 1937 Nobel Prize in Physics. (Note: G. P. Thompson was the
son of J. J. Thomson, Nobel-prize-winning discoverer of the electron.) Davisson’s Nobel lecture is very readable;
you can find a link to it in the Supplemental Material for Unit 12 in the Study Guide.
Given the above properties, we might think that de Broglie waves could be described by the form y(x,t)=Asin(kx-ωt).
But what physical variable does y represent here? It turns out that y is best viewed as a probability amplitude. Much
as the electromagnetic wave electric field does not give the intensity of the wave (but must be squared), the
probability amplitude must be squared to give the probability density of a particle. Integrating the probability density
over a region of space gives the probability of finding the particle in that region. The standard name for the variable
is actually wave function, often represented by the symbol ψ (psi). However, writing ψ(x,t)=Asin(kx-ωt) describes a
wave of infinite extent. Also, applying the energy relation E=ħω to particles would have to (for reasons beyond our
discussion in this course) take into account their total energy, which includes the rest mass, Einstein’s famous
E=mc2 term. For low-energy particles, we can avoid the discussion of special relativity (which is where the rest
mass formula comes from) and add the classical kinetic energy ½mv2 to the rest mass energy as the total energy. It is
also useful to use p=mv and write E    mc  ½ mv  mc 
2
2
velocity of de Broglie waves from general wave properties as v p 
2

k
p2
2m

1

. Then, we can calculate the phase
2
( mc 2  2pm )
p


mc 2
p , where the kinetic
energy term p2/2m was dropped, since for nonrelativistic (slow) particles it is much smaller than the rest mass
mc 2 c 2
v

 . Since the particle speed v is smaller than the
energy mc . If we now use p=mv again, we get p
mv
v
2
speed of light c, we have just found that the phase speed vp of de Broglie waves is faster than c! Despite
experimental proof (discussed above), the wave hypothesis for particles seems to have two big problems: 1) it does
not correspond to localized particles, and 2) the phase speed cannot correspond to motion of physical particles. We
must develop more knowledge of waves themselves before we can resolve these problems. Then we will see that the
de Broglie hypothesis is rich in results that correspond to reality. We will pursue these matters in a later unit.
Watch Video 1: Quantum Mechanics - Chapter 1 from the Cassiopeia Project.
Spectral analysis was developed in the 1750s. Scientists discovered that by placing substances in flames and passing
the light released through a prism, the colours of light spread out into spectra, and each spectrum was characteristic
of the substance involved.
In the video lectures, Walter Lewin demonstrates this effect with common table salt, producing a strong yellow
emission line from the sodium (Na) in the salt.
3 PHYS 302: Unit 8 Quantum Mechanics Supplement – Athabasca University
Spectra provided an excellent way to detect and identify substances, but how the spectra were produced was unclear.
The yellow-white light of the Sun (familiar to all) arises because the Sun is very hot and emits light at all visible
wavelengths. However, in 1802, studies with prisms (similar to what was used in the laboratory) showed gaps in the
Sun’s rainbow of colours. These gaps, called absorption lines, allowed for determination of the Sun’s composition
(although for various reasons, lines due to iron and other heavy elements are more dominant than the relatively small
proportion of those elements would lead one to suspect, initially). Still, the origin of the lines remained a mystery.
Solving this mystery, surprisingly, came from studying the continuous radiation from hot objects (which do not
show lines), rather than the direct study of spectral lines. In the ideal case, an emitter or absorber of radiation is
called a black body. Hot black bodies give off light whose wavelength grows shorter as the temperature increases.
A kiln with a small hole in its side is a good example of a black body for which the temperature can be accurately
controlled, and light observed through a small hole. In such a kiln, the electromagnetic radiation is in equilibrium
with the hot walls and forms a standing wave pattern with many wavelengths present. It was found, however, that as
the temperature went up, short wavelengths of light were present in lower amounts than classical theory would have
led scientists to expect: this situation was known as the ultraviolet catastrophe. Max Planck resolved this
uv catastrophe by suggesting that each colour of light was quantized due to the existence of photons that had to be
present in integral numbers. Under such a condition, the total would have been h ν. This also agreed with Einstein’s
interpretation that the photoelectric effect requires that photons have a certain maximum wavelength (or shorter) to
get rid of electrons.
The properties of photons are particle-like properties. Light has both wave and particle properties.
Watch Video 2: Quantum Mechanics - Chapter 2 from the Cassiopeia Project.
An electron moving in a circular orbit requires acceleration. This should cause light to be radiated, so that energy
would be released and the electron would spiral in to the nucleus. (Of course, this is not observed to happen;
undisturbed atoms can last “forever.”)
As in Planck’s case for energy of light, Bohr stipulated that only certain values of angular momentum (a particle-like
property) were possible for electrons in orbit, and claimed that such orbits were stable, though transitions were
possible between them. Energy differences in orbits were indeed found to match exactly. de Broglie (1924)
proposed that matter particles had wave-like properties. The orbits allowed had to be exact multiples of wavelengths
in circumference, to avoid destructive interference. The “orbits” were not orbits at all, but a standing wave pattern
with the exact position not specifiable at any given time.
Watch Video 3: Quantum Mechanics - Chapter 3 from the Cassiopeia Project.
Further developments in quantum mechanics brought about a probabilistic view. Werner Heisenberg stipulated that
the more precisely position is known, the less precisely momentum is known, and vice versa. This leads to a view in
which particles appear and reappear, with various probabilities of appearing at given locations. All possible paths
are sampled as particles move, and in some sense, the particles are “virtual.” At these tiny scales, existence of a
particle itself is inexact: “existence” is oscillating, and the wave is a map of the particle’s existence. Erwin
Schrödinger developed the equation that describes matter using probability waves. The solution of Schrödinger’s
equation is called the wave function; it reflects particle probabilities of existing matter in given regions.
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