Photoelectric effect
Emission of particles (electrons)
by a metal due to interaction
with light
Einstein’s explanation (for which
he got the Nobel Prize) involved
assuming that the energy of a
photon was proportional to its
frequency
Physical Chemistry
Lecture 11
Introduction to Quantum
Mechanics





Kinetic energy was independent
of intensity of light
Emission occurs only above a
threshold value
Kinetic energy is linear in the
frequency
Binding energy is known as the
work function
Results showed same
fundamental constant as Planck
K .E.  h
Spectra of atoms
Particles acting like waves
Early experiments



Louie de Broglie
proposed that particles
acted as waves in
certain situations
Observed only certain
emission frequencies
Spectra of atoms
Colors of mineral
solutions
Scientists involved
have very familiar
names







Blackbodies and Atomic Energy
Structure


E  h
Quantization of energy
h

Proposed the particle’s
momentum is related to
“wavelength”
Waves and particles
Waves


Blackbody radiation
Bohr’s rejection of
classical
electrodynamics to
interpret Rydberg’s
hydrogen-atom
spectrum
p 
Davisson and Germer
verified that electrons
diffract from a grating,
a wave-like behavior
Angstrom
Balmer
Bunsen
Fraunhofer
Kirchoff
Rydberg
Planck’s theory of the
quantization of modes
of a cavity
 

Delocalized structures
Stationary or time-varying
Structure depends on
matching conditions at
boundaries
Particles


1 
 1
 2
2
m 
n
~  RH 

Localized structures
Position is measurable
Classical mechanics
generally developed for
particles
1
Quantum mechanics
Quantum mechanical state
Conundrum


Waves sometimes act as if they are particles
Particles sometimes act as if they are waves
Quantum mechanics considers any object to be
neither and both


Particulate behavior occurs when the wavelength of the
particle is small compared to the dimensions of the space
Wave behavior occurs when the wavelength is comparable
to the dimensions of the space
Quantum mechanics is sometimes difficult to fathom
because



Most of our experience is with systems at the classical limit
It seems counterintuitive
It tends to abstract
With Heisenberg’s statement, there is no
longer a correspondence of position and
momentum to “state”
Definition of state of a quantum system


 Measurable properties
 Properties found by operation on the state function
Limited information available, consistent with
Heisenberg’s observation
Born’s idea: State is related to the probability of a
particular condition, since we cannot know properties
such as positions and momenta precisely.

Classical mechanics and states
In Newtonian mechanics,
one describes precisely


p  mv

t
p(t )  p(0) 
 F(t ' )dt '
0
r (t )  r (0) 
Hamilton’s mechanics
Equivalent to Newton’s
formulation
Based on energy, rather
than forces
Energy defined as the
Hamiltonian function
Where particles are
Where particles are going
subject to forces
The system’s state is
completely defined by the
positions and momenta of
all particles -- the
trajectories
Everything is absolutely
determined by forces and
initial conditions
t
1
p(t ' )dt '
m 0


Cannot know trajectories
exactly
Cannot describe a system
as precisely as with
classical mechanics
An inherent property of
all systems
Theory must reflect this
property
Must be a function of
position and momentum
Trajectories found from
operation on the
Hamiltonian

Heisenberg’s principle of
uncertainty
There is no way to
measure simultaneously
with infinite precision a
particle’s position and
conjugate momentum
The condition of a system
Gives all possible information
Hamilton’s canonical
equations are equivalent
to Newton’s equations of
motion
H
 T ( p)  V ( x)
T
V
 kinetic energy
 potential energy
H
p x

x
t
H
x
 
p x
t
Schroedinger’s equation
Primarily interested in constant-energy states


Start with Hamiltonian function of classical mechanics
Define operator for energy equivalent to the Hamiltonian
function
State specified by a wave function


Contains all information on a system’s state
Must solve for wave function
 Analogous to solving Newton’s equations or Hamilton’s
canonical equations in classical mechanics for trajectories
 Wave functions corresponding to constant qualities found
from eigenvalue equations

Standing-wave type functions
For constant-energy states
H

E
2
Quantum operators
To use Schroedinger’s
equation, must find
Hamiltonian operator


Must have operator for
momentum
Must have operator for
position
Use the Correspondence
Principle to get operators


Relates quantum operators
to classical variables
Direct substitution
Since there are derivative
operators in the
definitions, the eigenvalue
equations yield differential
equations
Hˆ
Tˆ



Tˆ

i
Vˆ

Vˆ
2
i
pˆ
2mi
V ( x)
pˆ x

 i
xˆ

x

x
Summary
Early experiments showed that many systems,
particularly small systems, only have certain
allowed energies
Experiments showed


Particles act like waves in some situations
Waves act like particles in some situations
Could reach agreement with observation by
“violating” classical mechanics -- Bohr theory
Uncertainty principle shows that small systems
could not be described by classical, deterministic
mechanics
Totally new theory



Definition of state
Operators for properties
Eigenvalue equations for property values
3