Failures of Classical Physics
Some experimental situations where "classical" physics fails:
Photoelectric effect
Blackbody radiation
Line spectra
Wave properties of electron
The remedies come from some inherently quantum ideas:
Photon energy
Photon momentum
Wavelength for
particle
Uncertainty principle
Wave function
Schrödinger equation
Blackbody Radiation
Blackbody radiation" or "cavity radiation" refers to an object or system which absorbs all radiation
incident upon it and re-radiates energy which is characteristic of this radiating system only, not
dependent upon the type of radiation which is incident upon it. The radiated energy can be considered
to be produced by standing wave or resonant modes of the cavity which is radiating.
The amount of radiation emitted
in a given frequency range
should be proportional to the
number of modes in that range.
The best of classical physics
suggested that all modes had an
equal chance of being produced,
and that the number of modes
went up proportional to the
square of the frequency.
But the predicted continual
increase in radiated energy with
frequency (dubbed the
"ultraviolet catastrophe") did not
happen. Nature knew better.
The birth of Quantum Mechanics
According to the Planck hypothesis, all electromagnetic radiation is quantized and occurs in finite
“quanta" of energy which we call photons. The quantum of energy for a photon is not Planck's
constant h itself, but the product of h and the frequency. The quantization implies that a photon of
blue light of given frequency or wavelength will always have the same size quantum of energy. For
example, a photon of blue light of wavelength 450 nm will always have 2.76 eV of energy. It occurs in
quantized chunks of 2.76 eV, and you can't have half a photon of blue light - it always occurs in
precisely the same sized energy chunks.
But the frequency available is continuous and has no upper or lower bound, so there is no finite lower
limit or upper limit on the possible energy of a photon. On the upper side, there are practical limits
because you have limited mechanisms for creating really high energy photons. Low energy photons
abound, but when you get below radio frequencies, the photon energies are so tiny compared to room
temperature thermal energy that you really never see them as distinct quantized entities - they are
swamped in the background. Another way to say it is that in the low frequency limits, things just blend
in with the classical treatment of things and a quantum treatment is not necessary.
Blackbody Intensity as a Function of Frequency
Energy per unit
volume per
unit frequency
k = Boltzman
constant
Rayleigh-Jeans vs Planck
Comparison of the classical Rayleigh-Jeans Law and the quantum Planck radiation
formula. Experiment confirms the Planck relationship
Radiation Curves
The Photoelectric Effect
The remarkable aspects of the
photoelectric effect when it was first
observed were:
1. The electrons were emitted
immediately - no time lag!
2. Increasing the intensity of the
light increased the number of
photoelectrons, but not their
maximum kinetic energy!
3. Red light will not cause the
ejection of electrons, no matter
what the intensity!
The details of the photoelectric effect
were in direct contradiction to the
expectations of very well developed
classical physics.
The explanation marked one of
the major steps toward quantum
theory.
4. A weak violet light will eject only
a few electrons, but their
maximum kinetic energies are
greater than those for intense light
of longer wavelengths!
The Photoelectric Effect
Analysis of data from the photoelectric
experiment showed that the energy of the
ejected electrons was proportional to the
frequency of the illuminating light. This
showed that whatever was knocking the
electrons out had an energy proportional to
light frequency. The remarkable fact that the
ejection energy was independent of the total
energy of illumination showed that the
interaction must be like that of a particle
which gave all of its energy to the electron!
This fit in well with Planck's hypothesis that
light in the blackbody radiation experiment
could exist only in discrete bundles with
energy
E  h
Most commonly observed phenomena with light can be explained by waves. But the
photoelectric effect suggested a particle nature for light.
The Line Spectrum Problem
In the years before the beginning of the 20th
century, the light emitted from luminous gases was
found to consist not of a continuous range of
wavelengths, but of discrete colours which were
different for different gases. These spectral "lines"
formed regular series and came to be interpreted
as transitions between atomic energy levels. This
presented a considerable problem for classical
physics, because accelerated charges were
known to radiate electromagnetic energy. It was
expected that orbits of electrons about positive
nuclei would be unstable because they would
radiate energy and therefore spiral into the
nucleus. No classical model could be found which
would yield stable electron orbits.
The Bohr model of the atom started the progress
toward a modern theory of the atom with its
postulate that angular momentum is quantized,
giving only specific allowed energies. Then the
development of the quantum theory and the
Schrödinger equation refined the picture of the
energy levels of atomic electrons.
Helium spectrum
Hydrogen spectrum
1
1 
 RH  2  2 

 n1 n2 
1
n1, n2
integers
(n1  n2 )
RH = Rydberg constant = 1.097 10–7 m–1
Bohr atomic model - Classical Electron Orbit
In the Bohr theory, this classical result was combined with the quantization of angular momentum to
get an expression for quantized energy levels.
Angular Momentum Quantization
In the Bohr model, the wavelength associated with the electron is given by the DeBroglie relationship
and the standing wave condition that circumference = whole number of wavelengths.
These can be combined to get an expression for the angular momentum of the electron in orbit.
Thus L is not only conserved, but constrained to discrete values by the quantum number n. This
quantization of angular momentum is a crucial result and can be used in determining the Bohr orbit
radii and Bohr energies.
Combining the energy of the classical electron orbit with the quantization of angular momentum, the
Bohr approach yields expressions for the electron orbit radii and energies:
substitution for r gives the Bohr energies and radii:
Hydrogen Energy Levels
The basic hydrogen energy level structure is in agreement with the Bohr model. Common pictures
are those of a shell structure with each main shell associated with a value of the principal quantum
number n.
This Bohr model picture of the orbits has some usefulness for visualization so long as it is realized that the
"orbits" and the "orbit radius" just represent the most probable values of a considerable range of values.
The Bohr model for an electron transition in hydrogen between quantized energy levels with different
quantum numbers n yields a photon by emission, with quantum energy
1
me4  1
1 
1 
h 
 2   13.6  2  2  eV
2 2  2
8 0 h  n1 n2 
 n1 n2 
This is often expressed in terms of the inverse wavelength or "wave number" as follows:
me4
RH 
2
8 0 ch3
RH  1.097 107 m1
Failures of the Bohr Model
While the Bohr model was a major step toward understanding the quantum theory of
the atom, it is not in fact a correct description of the nature of electron orbits. Some
of the shortcomings of the model are:
1. It fails to provide any understanding of why certain spectral lines are brighter than
others. There is no mechanism for the calculation of transition probabilities.
2. The Bohr model treats the electron as if it were a miniature planet, with definite
radius and momentum. This is in direct violation of the uncertainty principle which
dictates that position and momentum cannot be simultaneously determined.
The Bohr model gives us a basic conceptual model of electrons orbits and energies.
The precise details of spectra and charge distribution must be left to quantum
mechanical calculations, as with the Schrödinger equation.
Particle nature of light - Compton Scattering
Arthur H. Compton observed the scattering
of x-rays from electrons in a carbon target
and found scattered x-rays with a longer
wavelength than those incident upon the
target. The shift of the wavelength
increased with scattering angle according
to the Compton formula:
Compton explained and modeled the data
by assuming a particle (photon) nature for
light and applying conservation of energy
and conservation of momentum to the
collision between the photon and the
electron. The scattered photon has lower
energy and therefore a longer wavelength
according to the Planck relationship.
The above expression for Δλ can be obtained by exploting energy-momentum
conservation
Wave Nature of Electron
As a young student at the University of Paris, Louis DeBroglie had been impacted by relativity and the
photoelectric effect, both of which had been introduced in his lifetime. The photoelectric effect pointed
to the particle properties of light, which had been considered to be a wave phenomenon. He wondered
if electons and other "particles" might exhibit wave properties. The application of these two new ideas
to light pointed to an interesting possibility:
Confirmation of the DeBroglie hypothesis came in the Davisson- Germer experiment
which showed interference patterns – in agreement with DeBroglie wavelength – for the
scattering of electrons on nickel crystals.
When x-rays are scattered from a crystal lattice,
peaks of scattered intensity are observed which
correspond to the following conditions:
The angle of incidence = angle of scattering.
The pathlength difference is equal to an integer
number of wavelengths.
The condition for maximum intensity contained in
Bragg's law above allow us to calculate details about
the crystal structure, or if the crystal structure is
known, to determine the wavelength of the x-rays
incident upon the crystal.
The Davisson-Germer experiment
showed that electrons exhibit the
DeBroglie wavelength given by:
DeBroglie Wavelengths
Wave-Particle Duality: Light
Does light consist of particles or waves? When one focuses upon the different types of phenomena
observed with light, a strong case can be built for a wave picture:
Phenomenon
Can be explained in terms of
waves.
Can be explained in terms of
particles.
Reflection
Refraction
Interference
Diffraction
Polarization
Photoelectric effect
Compton scattering
Most commonly observed phenomena with light can be explained by waves. But the photoelectric
effect and the Compton scatering suggested a particle nature for light. Then electrons too were
found to exhibit dual natures.
Wavefunction Properties
Schrödinger Equation
The Schrödinger equation plays the role of Newton's laws and conservation of energy in classical
mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of
the wavefunction which predicts analytically and precisely the probability of events or outcome. The
detailed outcome is not strictly determined, but given a large number of events, the Schrödinger
equation will predict the distribution of results.
The kinetic and potential energies are transformed into the Hamiltonian which acts upon the
wavefunction to generate the evolution of the wavefunction in time and space. The Schrödinger
equation gives the quantized energies of the system and gives the form of the wavefunction so that
other properties may be calculated.
Time-independent Schrödinger Equation
For a generic potential energy U the 1-dimensional time-independent Schrodinger equation is
In three dimensions, it takes the form
for cartesian coordinates. This can be written in a more compact form by making use of the Laplacian
operator
The Schrodinger equation can then be written:
HΨ = EΨ
Time Dependent Schrödinger Equation
The time dependent Schrödinger equation for one spatial dimension is of the form
For a free particle where U(x) =0 the wavefunction solution can be put in the form of
a plane wave
k
2


2
p
h

2π
2
 2 
E
T
h
For other problems, the potential U(x) serves to set boundary conditions on the spatial part
of the wavefunction and it is helpful to separate the equation into the time-independent
Schrödinger equation and the relationship for time evolution of the wavefunction
The Postulates of Quantum Mechanics
1. The Wavefunction Postulate:
Associated with any particle moving in a conservative field of force is a wave
function which determines everything that can be known about the system.
It is one of the postulates of quantum mechanics that for a physical system consisting of a
particle there is an associated wavefunction. This wavefunction determines everything that
can be known about the system. The wavefunction may be a complex function, since it is
its product with its complex conjugate which specifies the real physical probability of
finding the particle in a particular state.
Probability in Quantum Mechanics
The wavefunction represents the probability amplitude for finding a particle at a given
point in space at a given time. The actual probability of finding the particle is given by
the product of the wavefunction with it's complex conjugate (like the square of the
amplitude for a complex function).
Since the probability must be = 1 for finding the particle somewhere, the wavefunction
must be normalized. That is, the sum of the probabilities for all of space must be equal
to one. This is expressed by the integral
*

  dV  1
dV  dx dy dz  infinitesimal volume
Part of a working solution to the Schrodinger equation is the normalization of the
solution to obtain the physically applicable probability amplitudes
2. The Operator Postulate
With every physical observable q there is associated an operator Q, which when
operating upon the wavefunction associated with a definite value of that observable
will yield that value times the wavefunction.
With every physical observable there is associated a mathematical operator which is used
in conjunction with the wavefunction. Suppose the wavefunction associated with a definite
quantized value (eigenvalue) of the observable is denoted by Ψn (an eigenfunction) and
the operator is denoted by Q. The action of the operator is given by
The mathematical operator Q extracts the observable value qn by operating upon the
wavefunction which represents that particular state of the system. This process has
implications about the nature of measurement in a quantum mechanical system. Any
wavefunction for the system can be represented as a linear combination of the
eigenfunctions Ψn (see basis set postulate), so the operator Q can be used to extract a
linear combination of eigenvalues multiplied by coefficients related to the probability of
their being observed (see expectation value postulate).
Operators in Quantum Mechanics
Associated with each measurable parameter in a physical system is a quantum mechanical operator.
Such operators arise because in quantum mechanics you are describing nature with waves (the
wavefunction) rather than with discrete particles whose motion and dynamics can be described with the
deterministic equations of Newtonian physics. Part of the development of quantum mechanics is the
establishment of the operators associated with the parameters needed to describe the system. Some of
those operators are listed below.
It is part of the basic structure of quantum mechanics that functions of position are unchanged in the
Schrödinger equation, while momenta take the form of spatial derivatives. The Hamiltonian operator
contains both time and space derivative.
3. Hermitian Property Postulate
Any operator Q associated with a physically measurable property q will be Hermitian.
The quantum mechanical operator Q associated with a measurable property q must be
Hermitian. Mathematically this property is defined by
*
*

(
Q

)
dV

(
Q

)
 a b
 a b dV
where Ψa and Ψb are arbitrary normalizable functions and the integration is over all of
space. Physically, the Hermitian property is necessary in order for the measured values
(eigenvalues) to be constrained to real numbers.
if Q is Hermitian, then all qi are real numbers
4. Basis Set Postulate
The set of eigenfunctions of Hermitian operators Q will form a complete set of
linearly independent functions.
The set of functions Ψj which are eigenfunctions of the eigenvalue equation
form a complete set of linearly independent functions. They can be said to form a basis
set in terms of which any wavefunction representing the system can be expressed:
This implies that any wavefunction Ψ representing a physical system can be expressed
as a linear combination of the eigenfunctions of any physical observable of the system.
5. Expectation Value Postulate
For a system described by a given wavefunction, the expectation value of any
property q can be found by performing the expectation value integral with respect
to that wavefunction.
For a physical system described by a wavefunction Ψ, the expectation value of any
physical observable q can be expressed in terms of the corresponding operator Q as
follows:
 q    *Q  dV
It is presumed here that the wavefunction is normalized and that the integration is over
all of space. This postulate follows along the lines of the operator postulate and the basis
set postulate. The function can be represented as a linear combination of eigenfunctions
of Q, and the results of the operation gives the physical values times a probability
coefficient. Since the wavefunction is normalized, the integral gives a weighted average
of the possible observable values.
A physical system is described by the wave function Ψ, which can always be
written as a linear combination of the eigenfunctions of a Hermitian operator Q:
   cn n
Q n  qn n
n
A measure of Q for the state Ψ will give as a result any of its eigenvalues qn,
each with a probability |cn|2, so that
 q  | cn | qn
2
n
The normalization condition of the wavefunction implies that
2
|
c
|
 n 1
n
A measurement of Q forces the system to be in one of the eigenstates, Ψn, of Q:
any subsequent measure of Q will give the result qn
6. Time Evolution Postulate
The time evolution of the wavefunction is given by the time dependent
Schrödinger equation
If Ψ(x,y,z; t) is the wavefunction for a physical system at an initial time and the system
is free of external interactions, then the evolution in time of the wavefunction is given by
where H is the Hamiltonian operator formed from the classical Hamiltonian by substituting
for the classical observables their corresponding quantum mechanical operators. The role
of the Hamiltonian in both space and time is contained in the Schrödinger equation.
1. Associated with any particle moving in a conservative field of force is a
wave function which determines everything that can be known about the
system.
2. With every physical observable q there is associated an operator Q,
which when operating upon the wavefunction associated with a definite
value of that observable will yield that value times the wavefunction.
3. Any operator Q associated with a physically measurable property q will
be Hermitian
4. The set of eigenfunctions of each Hermitian operator Q will form a
complete set of linearly independent functions.
5. For a system described by a given wavefunction, the expectation value
of any property q can be found by performing the expectation value integral
with respect to that wavefunction
6. The time evolution of the wavefunction is given by the time dependent
Schrödinger equation.
Free particle approach to the Schrödinger equation
Though the Schrodinger equation cannot be derived, it can be shown to be consistent with
experiment. The most valid test of a model is whether it faithfully describes the real world. The
wave nature of the electron has been clearly shown in experiments like the Davisson-Germer
experiment. This raises the question "What is the nature of the wave?". We reply, in retrospect,
that the wave is the wavefunction for the electron. Starting with the expression for a traveling
wave in one dimension, the connection can be made to the Schrödinger equation. This process
makes use of the deBroglie relationship between wavelength and momentum and the Planck
relationship between frequency and energy.

h
2
It is easier to show the relationship to the Schrödinger equation by generalizing this
wavefunction to a complex exponential form using the Euler relationship. This is the
standard form for the free particle wavefunction.
(ei  cos  i sin  )
One can check that Ψ is eigenunction of momentum and energy operators
The connection to the Schrodinger equation can be made by examining wave and particle
expressions for energy
Asserting the equivalence of these two expressions for energy and putting in the quantum
mechanical operators for both brings us to the Shrödinger equation
The Uncertainty Principle
The position and momentum of a particle cannot be simultaneously measured with
arbitrarily high precision. There is a minimum for the product of the uncertainties of these
two measurements. There is likewise a minimum for the product of the uncertainties of
the energy and time
This is not a statement about the inaccuracy of measurement instruments, nor a
reflection on the quality of experimental methods; it arises from the wave properties
inherent in the quantum mechanical description of nature. Even with perfect
instruments and technique, the uncertainty is inherent in the nature of things.
Uncertainty Principle
Important steps on the way to understanding the uncertainty principle are wave-particle duality and the
DeBroglie hypothesis. As you proceed downward in size to atomic dimensions, it is no longer valid to
consider a particle like a hard sphere, because the smaller the dimension, the more wave-like it
becomes. It no longer makes sense to say that you have precisely determined both the position and
momentum of such a particle.
The exact definition of Δx and Δp is:
x   x 2    x 2
p   p 2    p 2

x  p 
2
Particle Confinement
Confinement Calculation
The Hydrogen Atom
The solution of the Schrodinger equation for
the hydrogen atom is better achieved by
using spherical polar coordinates and by
separating the variables so that the
wavefunction is represented by the product:
The separation leads to three equations for
the three spatial variables, and their
solutions give rise to three quantum
numbers associated with the hydrogen
energy levels.
Quantum Numbers from Hydrogen Equations
The hydrogen atom solution requires finding solutions to the separated equations which obey
the constraints on the wavefunction. The solution to these equations can exist only when a
few constants which arise in the solution are restricted to integer values. This gives the
hydrogen atom quantum numbers:
H nlm  En nlm
En  
13.6
eV
2
n
n = principal quantum number
L2 nlm  l (l  1) 2 nlm
l = orbital quantum number
Lz nlm  ml  nlm
ml = magnetic quantum number
Vector Model for Orbital Angular Momentum
The orbital angular momentum for an atomic electron can
be visualized in terms of a vector model where the
angular momentum vector is seen as precessing about a
direction in space. While the angular momentum vector
has the magnitude shown, only a maximum of l units of ħ
can be measured along a given direction, where l is the
orbital quantum number.
While called a "vector", the orbital angular momentum in
Quantum Mechanics is a special kind of vector because its
projection along a direction in space is quantized to values
one unit of angular momentum (ħ) apart. The diagram
shows that the possible values for the "magnetic quantum
number" ml (for l =2), can take the values
= -2, -1, 0, 1, 2
Electron Spin
An electron spin s = 1/2 is an intrinsic property of
electrons. In addition to orbital angular momentum
electrons have intrinsic angular momentum
characterized by quantum number 1/2. In the pattern of
other quantized angular momenta
11 
S 2  spin    1  2  spin
22 
1
S z  spin  ms   spin     spin
2
ms= ½ “spin up”
ms= – ½ “spin down”
Spin "up" and "down" allows two electrons for each set of spatial quantum numbers
  nlml  ms
Pauli Exclusion Principle
No two electrons in an atom can have identical quantum numbers. This is an example
of a general principle which applies not only to electrons but also to other particles of
half-integer spin (fermions). It does not apply to particles of integer spin (bosons).
The nature of the Pauli exclusion principle can be illustrated by supposing that electrons
1 and 2 are in states a and b respectively. The wavefunction for the two electron system
would be
but this wavefunction is unacceptable
because the electrons are identical and
indistinguishable. To account for this we
must use a linear combination of the two
possibilities since the determination of
which electron is in which state is not
possible to determine.
The wavefunction for the state in which both states "a" and "b" are occupied
by the electrons can be written
The Pauli exclusion principle is part of one of our most basic observations of nature:
particles of half-integer spin must have antisymmetric wavefunctions, and particles of
integer spin must have symmetric wavefunctions. The minus sign in the above
relationship forces the wavefunction to vanish identically if both states are "a" or "b",
implying that it is impossible for both electrons to occupy the same state.
Pauli Exclusion Principle Applications
Periodic Table of the Elements
The quantum numbers associated with the atomic electrons along with the Pauli
exclusion principle provide insight into the building up of atomic structures and the
periodic properties observed.
For a given principal number n there are n2 different possible states.
The order of filling of electron energy states is dictated by energy, with the lowest
available state consistent with the Pauli principle being the next to be filled. The labeling
of the levels follows the scheme of the spectroscopic notation
Spectroscopic Notation
Before the nature of atomic electron states was clarified by the application of quantum
mechanics, spectroscopists saw evidence of distinctive series in the spectra of atoms
and assigned letters to the characteristic spectra. In terms of the quantum number
designations of electron states, the notation is as follows:
Order of Filling of Electron States
As the periodic table of the elements is built up by adding
the necessary electrons to match the atomic number, the
electrons will take the lowest energy consistent with the
Pauli exclusion principle. The maximum population of each
shell is determined by the quantum numbers and the
diagram at left is one way to illustrate the order of filling of
the electron energy states.
For a single electron, the energy is determined by the
principal quantum number n and that quantum number is
used to indicate the "shell" in which the electrons reside.
For a given shell in multi-electron atoms, those electrons
with lower orbital quantum number l will be lower in energy
because of greater penetration of the shielding cloud of
electrons in inner shells. These energy levels are specified
by the principal and orbital quantum numbers using the
spectroscopic notation. When you reach the 4s level, the
dependence upon orbital quantum number is so large that
the 4s is lower than the 3d. Although there are minor
exceptions, the level crossing follows the scheme indicated
in the diagram, with the arrows indicating the points at
which one moves to the next shell rather than proceeding to
higher orbital quantum number in the same shell
The division into main shells encourages a kind of "planetary model" for the
electrons, and while this is not at all accurate as a description of the electrons, it
has a certain mnemonic value for keeping track of the buildup of heavier elements.
The electron orbital configurations provide a structure for understanding chemical
reactions, which are guided by the principle of finding the lowest energy (most stable)
configuration of electrons. We say that sodium has a valence of +1 since it tends to
lose one electron, and chlorine has a valence of -1 since it has a tendency to gain
one electron. Both of these atoms are very active chemically, and their combination
is the classic case of an ionic bond.
Bose-Einstein Condensation
In 1924 Einstein pointed out that bosons could "condense" in unlimited
numbers into a single ground state since they are governed by Bose-Einstein
statistics and not constrained by the Pauli exclusion principle. Little notice was
taken of this curious possibility until the anomalous behavior of liquid helium
at low temperatures was studied carefully.
When helium is cooled to a critical temperature of 2.17 K, a remarkable
discontinuity in heat capacity occurs, the liquid density drops, and a fraction of
the liquid becomes a zero viscosity "superfluid". Superfluidity arises from the
fraction of helium atoms which has condensed to the lowest possible energy.
A condensation effect is also credited with producing superconductivity. In the
BCS Theory, pairs of electrons are coupled by lattice interactions, and the
pairs (called Cooper pairs) act like bosons and can condense into a state of
zero electrical resistance