Stefan constant is = 5.6703 10 8 2 4 , and is the absolute temperature in Kelvin. The The Wein displacement law says that ...

Phys 401
 Spring 2020
 Lecture #1 Summary
 27 January, 2020

 Welcome to Phys 401! It should be an exciting and action-packed semester as we delve
into the fascinating topic of Quantum Mechanics.

 We began by reviewing key ideas from Phys 371 “Modern Physics”. The first big step in
quantum mechanics came about from trying to understand the radiation emitted by blackbody
radiators. These are objects at some temperature that emit radiation over a broad range of
wavelengths, and their emission properties are independent of material. It was found empirically
that the total radiated power of such an object is given by the Stefan-Boltzmann law: =
 
 4 , where is the total radiated power per unit area of the object, having units of . The
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Stefan constant is = 5.6703 × 10−8 , and is the absolute temperature in Kelvin. The
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radiated power is spread out over a broad range of wavelengths as shown below. The wavelength
 of the peak radiated power is related to the temperature of the blackbody as ∝ 1/ .
The Wein displacement law says that = 2.898 × 10−3 − . This empirical result turned
out to be a key observation for the development of the photon theory of light.

 A blackbody radiator is realized by creating a box with interior walls at temperature . The
energy density of the radiation in the box is given by ( ), which has units of Energy/(Volume-

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Wavelength), or /( 3 − ). This is an energy density, both in terms of per unit volume, and
per unit wavelength. If there is a small hole in one wall of the box that lets some of the radiation
 
to escape, the radiated power at wavelength is given by ( ) = ( ), where is the speed of
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light. Hence the radiated power gives direct insight in to the energy density of radiation in the
box. Blackbody radiators are commercially available. The sun acts as a blackbody radiation
source to good approximation, at least when its ( ) spectrum is measured outside the atmosphere.

 The classical explanation was that ( ) = ( ) , where ( ) is the number of
electromagnetic modes with wavelengths between and + per unit volume inside the box.
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This quantity ( ) = is derived here. The factor of comes from the statistical mechanics
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idea of “equipartition of energy,” which says that every degree of freedom of the system acquires
 /2 of energy in equilibrium. This idea is very successful in the thermodynamics of ideal gases
 3
(where it predicts the internal energy = ), but it is a disaster for light in a box. It predicts
 2
the Rayleigh-Jeans law ( ) = 8 / 4 , which suffers from the “ultraviolet catastrophe” in that
it predicts an infinite energy density in the wavelength going to zero (ultraviolet) limit. This
contradicts the experimental situation, as shown in the figure above, where it is seen that ( ) is
strongly suppressed to zero at short wavelengths. Houston, we have a problem!

 To address this problem Planck made two (unjustified and revolutionary) assumptions:

 0) The atoms in the walls of the box have electrons, and when these electrons vibrate at
 frequency they emit electromagnetic waves at frequency . Likewise light at frequency
 can be absorbed by the atoms in the walls and start oscillating at frequency . In
 equilibrium there is an equal energy flux from the walls to radiation and from radiation
 back into the walls.
 1) The atoms in the walls of the box have discrete energy levels given by = , where
 = 0, 1, 2, … Hence the atoms can only interact with light of energy , 2 , 3 , etc.
 2) The energy of light is directly related to the frequency of oscillation of the EM fields as
 = ℎ , where ℎ is a fudge factor with units of / or − . Surprisingly, this energy is
 independent of the intensity of the light.

 Planck then adopted an assumption from statistical mechanics about the likelihood of finding
an atom in the walls of the box being excited to energy state , assuming the walls are in
equilibrium with the radiation field at temperature . It is given by the Maxwell-Boltzmann factor
at temperature : ( ) = − / , where is a normalization factor. Here ( ) is the fraction
of atoms in the wall of the box that are excited to a state of energy . Note the negative exponential
dependence on the ratio of the energy of the atom to the thermal energy . This means that
it will be very unlikely to find atoms occupying energy states with ≫ , which in turn means
that electromagnetic modes of the box at that energy (which corresponds to a short wavelength)
will have a very low probability of occupation, which will fix the ultraviolet catastrophe. The

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8 ℎ / 5
resulting Planck blackbody radiation formula is ( ) = ℎ / −1. This function has the property
that in the ultraviolet limit ( → 0) it goes to zero exponentially fast, thus avoiding the ultraviolet
catastrophe. To fit the blackbody emission data for ( ) one has to choose a fudge factor of ℎ =
6.626 × 10−34 − , which is known today as Planck’s constant.

 The photoelectric effect involves ultraviolet light impinging on a clean metal surface and
liberating photoelectrons. The parameters are the intensity and frequency of the light, and the
maximum kinetic energy of the photoelectrons, and the metal used for the photo-cathode. A
number of observations were made about this effect (see the experimental setup below):

 1) A positive anode (collector) voltage resulted in a steady photocurrent .
 2) The magnitude of the photocurrent scales with the light intensity ∝ .
 3) There was NO lag between the introduction of the light and the onset of photocurrent.
 Even in situations where the light intensity was so low that classically it would take
 hours to transfer enough energy to the electrons to begin liberating them from the metal,
 the first photoelectrons would appear essentially instantaneously after even weak light
 was turned on.
 4) If the anode (collector) potential was made sufficiently negative the photocurrent would
 cease. This is called the stopping potential − 0 (with 0 > 0).
 5) The stopping potential is a measure of the maximum kinetic energy of the liberated
 1
 electrons: 0 = � 2 �
 2 
 6) It was found that 0 is independent of the light intensity. Classically you would expect
 that higher light intensity would result in more energetic photoelectrons, but this is not
 observed.
 7) It was found that changing the frequency of the light would change the stopping
 potential. The higher the frequency of the light, the larger the magnitude of the
 stopping potential, and a linear (rather than quadratic) relationship was observed.

Observations 3, 6 and 7 are clearly at odds with the expectations of classical physics.

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Einstein’s 1905 paper originated the concept of a photon. He was bothered by the fact that
Maxwell’s equations predicted that light energy would continuously decrease to arbitrarily small
amounts as a light sphere around a source expanded outward. He proposed that this energy dilution
stopped when the light energy got down to some minimum quantum of energy. He also expected
that time-dependent phenomena involving electromagnetic waves (such as the absorption or
emission of light) might show new phenomena not described by Maxwell’s equations.

 Einstein made three proposals, with which he could explain all the experimental results on the
photoelectric effect:

 1) Adopt the energy quantization idea for electromagnetic fields, as proposed by Planck,
 namely the energy of the “light particles” is related to the frequency of the EM waves as
 ℎ = ℎ .
 2) The “quantum of light” aka “photon” collides with a single electron in the metal and
 transfers all of its energy to the electron at once.
 3) The energy of the resulting photoelectron is = ℎ − , where is the work function
 of the metal, and varies from one metal to the next, but is in the range of 2 to 5 eV. The
 work function is the binding energy of the electrons in the metal and depends on details.

 With this proposal, Einstein explained all the observations and made the following prediction.
In the limit of low frequency, the photon will not have enough energy to liberate the electrons
because ℎ < . This leads to the threshold frequency = /ℎ below which there is no
photocurrent . If one plots the stopping potential 0 vs. light frequency for > it should
obey the photoelectric equation: 0 = (ℎ/ ) − / , where the slope of the straight line should

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have a universal value of ℎ/ independent of the metal used in the cathode. This is in agreement
with experiments (see HW 1).

 Roentgen discovered in 1895 that X-rays are produced when cathode rays (electrons) are
produced with a very high potential (~ 104 Volts) and directed in to a metal target. The resulting
bremsstrahlung (braking radiation) is an electromagnetic wave. The wavelengths of the resulting
electromagnetic waves are less than 1 nm.

 X-ray emission has three properties:

 1) There is a continuous spectrum expected classically from “braking radiation.”
 2) There are sharp emission lines that appear for sufficiently high accelerating voltages.
 These lines depend on the type of metal used as a target.
 3) The continuous spectrum has a sharp cutoff at short wavelengths, found empirically to
 1240
 be described by the equation = − (Duane-Hunt Rule), where is the
 ( )
 accelerating voltage of the electrons.

 Einstein pointed out that X-ray generation is the inverse of photo-electron emission, and
 ℎ 1239.8
used the photoelectric equation to derive ≅ = − , in good agreement with the
 ( )
Duane-Hunt Rule. The observed sharp lines are a consequence of the discrete energy levels present
in atoms.

 Most solid materials are crystalline and are made up of atoms or molecules that are
regularly spaced in a periodic array. When x-rays come in to such a crystal structure they
encounter a periodic potential that can give rise to sharp and intense diffracted beams. These
beams arise from reflections of electromagnetic waves from parallel planes of atoms that act as
partially reflecting mirrors, and occur when constructive interference occurs in reflection. After
identifying a set of parallel planes, one can calculate the constructive interference diffraction
condition by requiring that each wave that penetrates one layer deeper must traverse a distance

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corresponding to an integer number ( ) of additional wavelengths before rejoining the beam
reflected from the layer above. This gives rise to the condition that = 2 sin , where =
1, 2, 3, …, is the wavelength of the x-rays, is the spacing of the parallel planes of atoms, and 
is the angle of incidence of the x-ray beam relative to the parallel planes. This is called Bragg’s
law, and is a consequence of the wave nature of x-rays.

 We briefly reviewed Compton scattering with an emphasis on the energy and momentum
of light particles (photons) involved in the scattering process. X-rays of wavelength scatter from
stationary electrons, changing to a longer wavelength ′ and moving off in a direction at angle 
 ℎ
relative to their incident direction. The Compton formula is: ′ − = (1 − cos ), where ℎ is
 
Planck’s constant, is the electron mass, and is the speed of light in vacuum. The idea that
light has an energy given by = ℎ , where is the frequency of oscillation of the electromagnetic
 
waves, and light has a momentum given by = = = = ℏ , is consistent with ideas
 
introduced by Planck and Einstein to explain the blackbody radiation spectrum and the photo-
electric effect, respectively. One derives the Compton formula simply by enforcing conservation
of energy and momentum for the “photon-electron particle-like collision.” This illustrates the
particle nature of light and further supports the concept of a photon. The fact that light sometimes
acts like a wave and sometimes acts like a particle is called “wave-particle duality.”

 The Nuclear Atom

 Atomic spectroscopy shows that atoms have many discrete spectral lines in emission and
absorption. Each type of atom has its own unique spectrum of lines, a kind of “fingerprint.” It
was found that Hydrogen (and one-electron ions) have the simplest structure of spectral lines.
Regularities in the spectral wavelengths were observed involving the inverse squares of integers.
For example the Rayleigh-Ritz formula ‘predicted’ the wavelengths of many spectral lines as
 1 1 1
 = � − �, where and are positive integers with > . The factor is the Rydberg
 2 2

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1
constant and has a value of = = 1.096776 × 107 for Hydrogen. Heavier single-electron
 
ions have slightly larger values of . Where does this regularity in the emission spectrum come
from?

 Rutherford scattering (studied carefully in Phys410 Classical Physics) experiments consist
of energetic α-particles (2 protons plus 2 neutrons) being scattered from materials like Au in the
form of a thin sheet. It was found that a large number of the α-particles were scattered straight
back, and the angular distribution of the scattering implies that much of the mass of the atom is
concentrated in a single positively charged entity. This entity is the nucleus of course, and
Rutherford showed that it has a dimension on the scale of 1 fm (10−15 m). The electrons are
distributed more or less uniformly in a cloud outside of the nucleus. This is the origin of the
nuclear model of the atom.

 Neils Bohr developed a planetary model of the Hydrogen atom. One can imagine an
electron in an orbit around the proton in analogy with the earth around the sun. However, in
classical physics charges that accelerate are known to radiate. Thus it was not clear why an
electron in an orbit around the proton would not radiate continuously and spiral into the nucleus.
Bohr made three bold postulates:

 1) Electrons orbit the nucleus in circular orbits called “stationary states” and do not radiate
 while in such states.
 2) Atoms radiate when electrons make transitions between stationary states of different
 energy.
 3) The angular momentum of electrons in stationary states is quantized in units of ℏ = ℎ/2 .
 (Here, Planck’s constant appears in an entirely new context, very different from blackbody
 radiation!)

 Bohr then did a “semi-classical” calculation of the Hydrogen atom structure, assuming that the
electron and proton are attracted to each other by the Coulomb interaction. He arrived at the
following results:

 Using Newton’s second law of motion for the bound state of a positively charged nucleus and
a single negatively charged electron, he found the speed, angular momentum, and the radius of the
electron orbit, and the total energy of the Hydrogen atom. The speed of the electron in its circular
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orbit of radius : = �4 , where the nucleus has charge + , 0 is the permittivity of free
 0 

space, and is the (reduced) mass of the electron.

 Now use the third postulate and assume a non-relativistic situation: The angular momentum is
quantized as � �⃗� = | ⃗ × ⃗| = = ℏ, with = 1, 2, 3, … This leads to the quantized radii
 2 0 4 0 ℏ2
of the Bohr orbits: = , where 0 = = 0.529 Å is called the Bohr radius. Note that
 2

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the Bohr radius expression is made up of only fundamental constants of nature. Hence the structure
of the Hydrogen atom is universal and time invariant, as far as we know.

 The total energy of the hydrogen atom = + ( is kinetic energy of the electron – the
proton is assumed to be stationary, is the Coulomb potential energy between the electron and
 2
 2 2 � 2 /4 0 � 2
proton) is found to be = − 0 2 , with 0 = 2 (ℏ )2
 = 2 = 13.6 , where =
 2
 2 /4 0 1
 ≅ is a famous dimensionless constant called the fine structure constant. For
 ℏ 137.036
 13.6 2
Hydrogenic atoms and single-electron ions one has = − , with = 1, 2, 3, … Why is
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the energy negative? Negative energy represents a bound state of the electron and proton. We
define the zero of energy to be an electron and proton separated to infinity, with no kinetic energy.
A negative energy comes about because the Coulomb attraction of the two particles, plus their
kinetic energy, corresponds to a lower energy state than having the particles at rest separated to
infinity. Also take note that Bohr predicts an infinite number of bound states of a proton and
electron (labelled by the set of all positive integers)!

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