Chemistry 4521
Time is flying by: only 15 lectures left!!
 Six quantum mechanics
 Four Spectroscopy
 Third Hour exam
 Three statistical mechanics
 Review
 Final Exam
Exam, Wednesday
Wednesday, May 4
4, 7:30 – 10 PM
Quantum Mechanics
Overall goals:
Introduce QM and qualitatively solve limited problems in 1D; extend to 3D
Particle in a box (translation quantization)
Harmonic oscillator
ll
(molecular
( l
l vibration)
b
)
Rigid rotor (molecular rotation)
Atomic and molecular electronic energy levels
Apply these results to spectroscopic analyses and statistical equilibrium
Lecture 1 (today)
 Classical mechanics, optics,
p
wave motion, thermodynamics
y
Failures when extending to short distances, small masses
Blackbody radiation ( existence of photons)
Wave –particle duality
Diffraction,
Diff
ti
photoelectric
h t l t i effect
ff t
Matter (deBroglie) waves, electron diffraction
Qualitative QM
Bohr atom
Bohr
Particle in a box
All were partial answers, leading Schrödinger to wave mechanics
Classical Physics/Mechanics
 Predicts precise trajectory for particles with precisely
specified
ifi d llocations
ti
and
d momenta
t att each
h instant
i t t
 Allows the translational, rotational and vibrational
modes of motion to be excited to any energy, simply
by controlling forces that are applied
 Considers matter and energy as distinct
 Shattered by three observations involving matter and
light, all indicating the presence of discrete energy states

• Blackbody radiation
• Photoelectric
Ph t l t i effect
ff t
• Atomic line spectra
Quantum Mechanics
Circumference = 2r=n
r

Describes the "Wave-Particle" Duality
Light is an electromagnetic wave,
wave described by Maxwell
Maxwell’ss equation
- but it can also behave like a particle
Particles - also have wavelike nature (manifest only when mass is tiny)
The wave properties
Th
ti of
f particles
ti l can be
b described
d
ib d by
b a
modified form of Maxwell’s equations for wave motion, known
as the Schrödinger equation.
Pre-Quantum Mechanics
1890's
I. Classical Mechanics
General Equations
q
(F
(F= m
ma on steroids))
LaGrange
Hamilton
II. Electricity & Magnetism
II
Maxwell's Equations
Electromagnetic Waves:
Central theses of the time: No real conceptual issues remain unresolved.
C
Computations
t ti
on reall systems
t
were unbelievably
b li
bl hard,
h d however!
h
!
Traveling
g Wave
Wave amplitude is orthogonal
to the direction of propagation
Single vertical pulse moves on horizontal string
With only five lectures, we must ignore time
dependence and restrict o
dependence,
ourselves
rsel es to those
states characterized by standing waves.
Time-independent quantum mechanics
 2
Y  x ,t   A sin 


 x  vt 
with frequency  
v


Wave Motion in Restricted Systems
y
One-half
wavelength
(/2) is
i the
h
“quantum”
of the
guitar
string’s
vibration
X
X
Pre-Quantum Mechanics
II. Electricity & Magnetism
Maxwell's Equations
Light:
III. Thermodynamics
(you know all about this!)
IV. Optics
Wave Diffraction
( ~ object size)
Two slit diffraction:
Geometrical Optics
( >> object size)
Failures of Classical Physics
(waves behaving as particles)
1. Blackbody radiation
Bl kb d R
Blackbody
Radiation
di ti
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Smoldering coal
1000 K
Electric heating coil
1500 K
Light bulb filament
2000 K
Radiation from a Cavity
i Blackbody
is
Bl kb d R
Radiation
di ti
Atoms in solid vibrate
and produce radiation
Radiation
a at on in
n cavity
ca ty has
range of frequencies, 
Treated rigorously
with classical
thermodynamics
and E & M
Rayleigh-Jeans
y g
Blackbody
y Calculation
(rigorous thermo and E&M)
8
  3 kBT d
c
8
  4 kBT d
2

H
How
well
ll did it work?
k?
The “Ultraviolet Catastrophe”
Correct at
large 
Blackbody
y Radiation
Radiation emitted from object increases and shifts to
shorter wavelengths at higher temperatures
Not quantitatively explained by classical physics
 is
i energy att a frequency
f
, per unit volume, and
per unit frequency range
Planck's Idea that Oscillator
Energy is Quantized
Max Planck "fixed it up" by trial and error, in the process
incorporating
p
g a "non-physical"
p y
assumption:
p
Photons can behave like particles under some conditions, and
have a relationship between their frequency and energy given by
E = h  = h c/, where h is a constant.
Substituting
S
b i i this
hi h
hypothesis
h i iinto the
h Rayleigh
R l i h - Jeans
J
theory,
h
he
h
obtained
8 hc  1 

 (T ,  )  5  hc
  kT 
 e  1
Planck simply adjusted h
h, and found one value
value, 6
6.6
6 x 10-277 erg s,
s
that fit the experimental data!
It was later known as Planck's constant.
No one took it seriously.
Comparison of Planck and Rayleigh-Jeans
equations
R l i h Jeans
Rayleigh
J
(non-quantized
(
i d energy):
)
8
  3 kBT
c
2
Pl
Planck
k (quantized
(
ti d energy):
)
8
h
  3
c exp(h / kBT)  1
2
A constant (later called h) reappeared in the photoelectric
effect and in the diffraction of “matter waves,”
in each case with the same numerical value! A clue!
Failures of Classical Physics
(waves behaving as particles)
2. Photoelectric effect
Light
Electron
Photoelectric Effect
Light
Electron,
mass me
speed u
Electron emission depends on frequency of light,
NOT on intensity of light
Light acts as if it is a beam of particles that has energy h
p between photon
p
absorbed and e– emitted
m
One-to-one relationship
h = 1/2 mu2 + w
where w = e= work
k function (E required to remove e–)
and 1/2 meu2 = KE of ejected e–
Experimental Characteristics of the
Photoelectric Effect
h = 1/2 mu2 + e


1. There is a threshold frequency for
electron ejection. Neither the threshold  nor
KE depends on intensity of the light.
light
2. KE of ejected electrons increases with
frequency .
3. There is no time lag for electron ejection

4. e is the “work function” (e- binding
energy) of the metal
5. Nobel prize for Einstein!

ConcepTest # 1
The kinetic energy
y of the photop
electron is plotted versus the
frequency of incident radiation for
potassium, rubidium and sodium.
F
From
left
l ft to
t right,
i ht identify
id tif the
th
lines.
A. K, Rb, Na
B Rb,
B.
Rb Na,
Na K
C. Rb, K, Na
D Na,
D.
N K,
K Rb
Failures of Classical Physics
(particles behaving as waves)
3. X-ray and Electron Diffraction
d
 x ray
d
e
m
e-
Led to deBroglie

V
d
Au
matter 
h
pmatter
p = momentum = mv
Au
4. Atomic Line Emission seems to indicate discrete energy states
Dispersion of visible light
Dispersion
p
off light
g ffrom excited atoms
Photons as waves and particles
Light particles with energy h known as
photons
Dual theory
y of radiation
Diffraction and interference depends on
WAVE properties of light
Photoelectric effect depends on
PARTICLE properties of light
Matter: particles & waves
Matter with mass m and velocity v seems to have a
wave of wavelength
g D = h/p
p associated with it.
Matter behaves “normally” (F=ma) when it interacts
with objects with dimensions >> D
Matter displays diffraction and interference
effects when it interacts with objects with
di
dimensions
i

D.
(constructive and destructive interference leading
to the concept
p of stationary
y states))
h
D 
p
The Schrödinger equation is essentially a
wave equation applied to matter waves.