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PHYSICAL CHEMISTRY II
CHEM 3354
• Topics: Quantum mechanics, spectroscopy, and statistical
mechanics
• Grading: Three exams worth 75%, ACS certified
comprehensive (over all p.chem) final worth 25%.
• You are strongly encouraged to read “In Search of
Schrodinger’s Cat” by John Gribbon.
The Origins of Quantum Mechanics
• Classical mechanics was introduced in the 17th century by
Isaac Newton.
– Successful at explaining the motion of everyday objects and planets
– Fails when applied to very small particles.
– The failures of classical physics lead to the discovery of quantum
mechanics.
• Quantum mechanics was discovered in the 1920s to describe
very small particles.
– The basis of quantum mechanics is that energy is quantized – has
discreet values.
– Particles have wave-like characteristics.
Black Body Radiation
• Hot objects emit electromagnetic radiation.
– Example: Hot objects glow “red hot”.
– Hotter objects glow “white hot”.
• We’ll soon see that we need quantum mechanics to describe
this phenomena.
Black body radiation
• These are the curves from an ideal
black body.
– An ideal blackbody is an object
capable of emitting and absorbing
all frequencies of radiation
uniformly.
• At lower temperature the
wavelength with maximum
intensity is toward the red end of
the spectrum.
• As T increases, the wavelength of
maximum intensity blue shifts.
– But there’s still a lot of intensity
near the red end of the spectrum.
– This is why hotter things appears
white.
Ideal Black Body
• A pinhole in an enclosed container.
– Any emitted radiation will be reflected MANY times
before leaving through the hole.
– The emitted light will consequently be in thermal
equilibrium with the wall at temperature T.
Wien Displacement law
• Predicts the wavelength of
maximum emission at a
temperature T.
• Based only on experimental
observation. (i.e. empirical)
Stefan-Boltzmann Law
• Describes the total energy
density (the area under the
curve)
– Total energy density - the
total electromagnetic energy
in a region divided by the
volume.
– Stefan derived the formula
from experimental
observations.
– Boltzmann showed the
formula theoretically.
Rayleigh-Jeans Law
• Nineteenth century scientists had
a difficult time explaining blackbody radiation.
• Lord Rayleigh & James Jeans
derived from the laws of classical
physics the Rayleigh-Jeans law:
– The electromagnetic field was
believed to be a collection of
oscillators of all possible
frequencies.
– The presence of radiation of
frequency n signified that the
oscillator of that frequency had
been excited.
– Average energy of each oscillator
was kT from equipartition
principle.
Electromagnetic vacuum supports
oscillations of the EM field
Rayleigh-Jeans law
• According to classical physics,
even cool objects should radiate
in the visibile and UV regions.
– In other words, objects should
glow in the dark.
– Wein and Stefan-Boltzmann laws
were not predicted.
• This break-down of classical
physics is known as the UV
catastrophe – first observed
failure of classical mechanics.
• Why did it fail?
– Rayleigh assumed that the
electromagnetic energy varied
uniformly.
Planck’s Distribution
• Planck proposed that the electromagnetic energy was limited
to discrete values, quantization of energy.
• Applying this assumption and thermodyanamics, Planck
derived the Planck Distribution:
Planck’s Distribution
• Matches experimental data by
predicting that cold object will not
emit high energy light.
• First evidence that energy is
quantized.
• Planck’s distribution could also
account for Stefan-Boltzmann and
Wien laws.
Energy Quantization and Thermal Properties of
Solids
• It was shown by Einstein that energy quantization must be
introduced in order to explain the thermal properties (heat
capacity) of solids.
Correct shape
Poor fit to data
Debye improved on Einstein’s work
Atomic Spectra and Energy Quantization
• Atomic and molecular spectra provide the most compelling
evidence for the quantization of energy.
– Energy is discarded or absorbed only in discrete amounts.
Atomic Spectra and Energy Quantization
Explanation of Spectral Lines
• Spectral lines appear because the
molecule emits a photon as it
changes between discrete energy
levels.
– High-frequecy radiation is
emitted when the energy change
is large.
– DE = h n
CONCLUSION
Energy is quantized.
Wave-particle Duality
• Classical physics treats electromagnetic radiation as a wave.
• Classical physics treats electrons as particles.
• Let’s look at some experiments that invalidate these ideas.
The Particle Character of EM Radiation
Photo-electric effect
•
•
Electrons are ejected from metals
when exposed to ultraviolet radiation.
Experimental Observations:
1. No electrons are ejected, regardless
of intensity, unless its frequency
exceeds a threshold value
characteristic of the metal.
2. The KE of the ejected electrons
increases linearly with the frequency
of the incident radiation but is
independent of the intensity of the
radiation.
3. Even low intensities immediately
eject electrons if the frequency is
above threshold.
•
Conclusion: the ejection of an
electron occurs due to a collision
with a particle-like projectile that
carries enough energy to eject the
electron from the metal.
Photo-electric effect
• Light behaves as a particle, called
photons, each photon has an
energy
E  hv 
hc

Photon energy is
insufficient
Photon energy is
sufficient and excess
energy is carried
away as KE of the
photoelectron
Davisson-Germer Experiment
• The scattering of an electron
beam from a nickel crystal
showed a variation of intensity.
– This variation is characteristic of
a diffraction experiment in which
waves interfere constructively
and destructively in different
directions.
• The experiment has been repeated
with heavier particles (including
molecular hydrogen) and shows
clearly that particles have wavelike properties.
• Electron diffraction the basis of
electron microscopy.
– Analogous to optical microscopy
but the image is constructed from
the pattern of scattering of a
focused beam of electrons.
The de Broglie wavelength
•
De Broglie proposed that any moving
particle (with a linear momentum r)
should a wavelength
– These “fictitious waves” are called
mater waves.
– The de Broglie relation can be
derived from Eistein’s relativity
theory and the Planck-Einstein
relation.
– The de Broglie relation can also be
derived by creating a standing wave
on a ring (i.e. Bohr atom and
quantization of angular momentum).
•
Exercise: Calculate the de Broglie
wavelength of a baseball of mass 5.1
ounces thrown at 59 miles/hour.
– Macroscopic bodies have such high
momenta (due to large m) that their
wavelengths are undetectably small.
Conclusion
Matter and light can behave as particles or waves.
A wave-particle duality exists.
Classical physics treats particles and waves as
separate entities. Must have a new
description to account for this mysterious
behavior.
Brace yourselves for quantum mechanics!!!!
Units and constants in Quantum Mechanics
• Scientists have developed a set of energy units that are
convenient for describing atomic and molecular energies.
• Energy Units
– Electron volt = eV = 1.602 x 10-19 J {electron times a volt}
– Hartree = Eh = 27.2 eV
– Wavenumber = cm-1 with 1 eV = 8065.5 cm-1; 1 cm-1 = 1.9864 J
• Commonly used constants.
–
–
–
–
–
Speed of light = c = 2.998 x 108 m/s
Planck’s constant = h = 6.626 x 10–34 J s
Atomic mass unit = u = 1.66054 x 10-27 kg
Angstrom = Å = 10-10 m
Bohr = ao = 0.529 Å
Classical Wave Theory
Physics Review
• Consider the harmonic-motion of a one-dimensional string (i.e.
a guitar string)
– The classical time-independent wave equation is:
– The solution in 1-D is: 3-D:
– Apply results to particle with a de Broglie wavelength .
Time Independent Schrödinger Wave
Equation (SWE)
Hˆ   E
2

Hˆ  
2  V
2m
• This is postulate of QM, accounts for wave character of matter.
• The wavefunction y contains all the dynamical information
about the system it describes.
– The trick is to determine what y is.
– And to figure out how to extract the desired information.
Interpreting the wavefunction
Born interpretation of y
• According to the wave theory of light, the square of the
amplitude of an EM wave is proportional to the intensity of
light.
– But since light behaves as a particle, the intensity must be a measure of
the probability density of photons in a volume of space.
• Applying this same idea to particles indicates that the value of
|y|2 at a point is proportional to the probability of finding the
particle at that point.
Born Interpretation of y
• The probability of finding a
particle between x and x+dx is
proportional to |y|2dx.
|y|2  probability density (real
and never negative).
– y  probability amplitude
–
• Can be complex
In 3-D, the probability of finding
a particle in an infinitesimal
volume dt = dxdydz is
proportional to |Y|2 dt
The Born interpretation places some
restrictions on the form of the wavefunction.
1.
2.
3.
4.
5.
y must be finite everywhere; otherwise
couldn’t be normalized.
y must be single valued; otherwise could
have more than one probability.
y must be defined everywhere.
The second derivative of y must be welldefined; or the SWE will not be applicable
everywhere.
These restrictions prevent the possibility of
finding an acceptable solution to the SWE
for arbitrary values of E.
–
Not continuous
Slope is discontinuous
Not single-valued.
Only certain energies (quantization) have
acceptable solutions.
Infinite over a finite region.
The Born Interpretation and Normalization
• The probability that a particle is somewhere must be 1.
– For a normalized wave function in 1-D:

*

 ( x) ( x)dx  1

– If the wavefunction is NOT normalized, we multiply y by a constant
factor N to normalize it.

N
2
*

  dt  1

Normalization Example
• The wavefunction for an electron in the
lowest energy state of a hydrogen atom is
proportional to e-2r/ao where ao is a constant
(the bohr radius) and r is the distance the
electron is from the nucleus. Normalize this Spherical Coordinates
wavefunction.
How do we get other information (besides
position) out of the wavefunction?
Hˆ   E
Hˆ  Hamiltonia n operator, energy operator - extracts the energy from  .
• The Schrodinger equation is an eigenvalue equation!
(operator)(eigenfunction) = (eigenvalue)(same eigenfunction)
• Examples
– Are these functions eigenfunctions of the d/dx operator? If so, what are the
eigenvalues.
e ax
e
ax 2
– Is the function cos (ax) an eigenfunction of:
d
dx
d2
dx 2
To ‘solve the Schrodinger equation’ is to ‘find
the eigenvalues and eigenfunctions of the
Hamiltonian operator for the system of
interest.’
•The wavefunctions are the eigenfunctions of Ĥ.
•The corresponding eigenvalues are the allowed energies.
Let’s solve it…
• Solve the SWE for a particle with mass m free to move on the
x-axis w/ zero potential energy (V = 0).
– cos(kx) to is an eigenfunction of the SWE (and consequently a valid
solution) with an eigenvalue.
 2k 2 p 2
E

; p  k
2m 2m
• What is k?
– It’s the “quantum number”. BUT since the solution is valid for ANY
value of k the energy of this particle is NOT quantized.
– k controls the frequency of the cosine wave.
• Large k  high momentum  many oscillations  wiggly wavefunction
 large curvature.
• Small k  low momentum  few oscillations  smoother wavefunction
 small curvature.
The Momentum Operator
• There’s a QM operator for all classical mechanical
observables.
– Ĥ is the energy operator  extracts the energy from Y.
• Finding the momentum from this energy only gives its magnitude, not its
direction.
 2k 2 p 2
E

; p  k
2m 2m
– The momentum operator is.
pˆ x 
 d
i dx
• Problem: Doesn’t work with cos (kx), not an eigenvalue equation.
Solution
• Let’s represent cos(kx) in terms of exponential functions, a more flexible
solution.
• A wavefunction that is not an eigenfunction of the operator of interest
can be written as a linear combination of eigenfunctions.
• Conclusion: The wavefunction is a superposition of states (moving
EITHER left OR right with momentum ħk).
– We don’t know which state our system is in until we observe it.
– Since both directions are equally probable, we cannot predict the direction the
particle will be traveling.
– Introduce the Schrodinger cat paradox.
Schrödinger’s Cat Paradox
• Cat is in closed container containing a vial of hydrocyanic acid.
• One atom of a radioactive substance (half life = 1 hour) is in a device with
a Geiger counter.
• If the Geiger counter detects a decay, it triggers the release a hammer to fall
and break the vial—killing the cat.
• Since the probability of decay is 50/50, there is a 50/50 chance the cat will
be alive after 1 hour.
• The cat is both dead and alive until it is observed.
For all classical mechanical observables, there’s a
corresponding quantum mechanical observable.
For all classical mechanical observables, there’s a
corresponding quantum mechanical observable.
• If y is an eigenfunction of our quantum mechanical operator,
then we can operate on y to obtain our classical mechanical
observable.
• What if y is NOT an eigenfunction of our operator?
– Then we can calculate an expectation value of this operator.
• An expectation value is a weighted average of a large # of observations of
a property.
 Oˆ    *Oˆ dt
NOTE :  must be normalized .
Example expectation value problem
• Calculate the average value of the distance of an electron from
the nucleus in the hydrogen atom.
PROOF
• A wavefunction that is not an eigenfunction of the operator of
interest can be written as a linear combination of
eigenfunctions.
– Example: cos kx was not an eigenfunction of the momentum operator
but eikx + e-ikx was.
– Find <Ô> for Y = c1f1 + c2f2 where f1 and f2 are eigenfunction of Ô.
• The expectation value is the sum of the eigenvalues weighted
by the probabilities that each one will be found in a series of
measurements.
Heisenberg’s uncertainty principle
• For y = Aeikx we found that the particle was moving to the right with px =
+kħ.
– Therefore, we know its momentum exactly.
• Where is the particle? (probability of finding the particle at a location x)
– Probability = Y* Y = A2 e-ikx eikx = A2
– There is an equal probability of finding the particle anywhere on the x-axis.
• Conclusion:
– We know the momentum of the particle exactly.
– BUT, we know NOTHING about the location of the particle.
• It is impossible to specify simultaneously both the momentum and position
of a particle.
Heisenberg’s uncertainty principle
• If we know a particle is in a
definite location, then y must be
large there and zero everywhere
else.
Heisenberg’s uncertainty principle
• To make such a wavefunction
(and still have a solution to the
SWE) it must be a linear
combination of many harmonic
(sine or cosine) functions, or
equivalent, eikx functions.
– It takes an infinite # of these to
make an infinitely narrow spike.
– But each eikx function
corresponds to a different linear
momenta.
– Therefore there’s an infinite
number of momenta possibilities.
• It is impossible to specify
simultaneously both the
momentum and position of a
particle.
Heisenberg’s uncertainty principle
• The quantitative interpretation of the uncertainty principle is:
Dpx Dx  
Uncertainty in
the momentum
2
Uncertainty in
the position
Dp and Dx are the
‘uncertainties’ in the
momentum and position,
respectively.
Dp 
p  p
D
x

x

x
approach zero, the other observable must approach infinity.
2
As either Dx or Dpx
2
2
2
Heisenberg’s uncertainty principle and
operators
• Heisenberg’s uncertainty principle applies to any pair of
operators  and B that do NOT commute.
– The commutator of two operators is defined by:
[ Aˆ , Bˆ ]  Aˆ Bˆ  Bˆ Aˆ
– If [Â ,B] = 0, then the operators commute and CAN be measured
simultaneously.
• Find the commutator of px and x.
• Find the commutator of px and Ť.
– What does this tell you about these two observables?