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Quantum Mechanics:
Wave Theory of Particles
Dr. Bill Pezzaglia
QM Part 2
Updated: 2010May11
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Quantum Mechanics
A. Bohr Model of Atom
B. Wave Nature of Particles
C. Schrodinger Wave Equation
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A. Bohr Model of Atom
1. Bohr’s First Postulate
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Electron orbits are quantized by
angular momentum
Orbits are stable, and contrary to
classical physics, do not
continuously radiate
Principle Quantum number “n”
(an integer whose lowest value is
n=1)
Niels Bohr
1885-1962
1922 Nobel Prize
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1. Bohr’s First Postulate
(a) Quantized Angular Momentum
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1912 first ideas by J.W. Nicholson
Postulates angular momentum of electron in
atom must be a multiple of
h

L  mvr  n
2
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1. Bohr’s First Postulate
(b) Stationary Orbits
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Classical physics says accelerating charges
(i.e. electrons in circular orbits) should radiate
energy away, hence orbits decay.
•
Bohr says orbits are stable and do not radiate
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Principle quantum number “n” has a lowest
value of n=1 (lowest angular momentum of one
h-bar).
(c) The Bohr Radius
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Classical equation of motion
Substitute:
Solve for radius:
Bohr Radius:
v
( Ze)e
m 
2
r 40 r
L
n
v

mr mr
a0
rn  n
Z
2
h 0
a0 
 0.053 nm
2
me
2
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2. Bohr’s Second Postulate
(a) The sudden transition of the electron
between two stationary states will produce
an emission (or absorption) of radiation
(photon) of frequency given by the
Einstein/Planck formula
hf  Ei  E f
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(b) Energy of nth orbit
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Viral Theorem: For
inverse square law force:
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Hence total energy:
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KE   12 PE
E  KE  PE  12 PE
Use Electrostatic energy
formula, we get:
2
Ze
E
80 r
(b) Energy of nth orbit
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Substitute Bohr’s radius formula for n-th
orbit gives energy of nth orbit:
2
2
Z hcR
Z
En  
 (13.6 ev) 2
2
n
n
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Where he can calculate Rydberg’s
constant from scratch!
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e
me
R
 3 2
80 hca0 8h  0 c
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(c) Bohr Derives Balmer’s Formula
•
From Einstein-Planck Formula:
hf 
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hc

 Ei  E f
Substituting his energy formula (and
divide out factor of hc), he derives
Balmer’s formula!
1
1 
 Z R 2  2 

n f 
 ni
1
2
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3. Bohr’s Correspondence Principle
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1923: Classical mechanics “corresponds” to
quantum system for BIG quantum numbers.
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When “n” is big, it behaves classically
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When “n” is small, it behaves “quantumly” (is
that a word?)
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B. Wave Nature of Particles
1. deBroglie Waves
2. Particle in a Box
3. Heisenberg Uncertainty
1. deBroglie Waves (1924)
a)
Suggest particles have wavelike
properties following same rules as
photon.
Ehf
h
P

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Proof: 1927 Electron diffraction
experiment of Davisson & Germer
(Nobel Prize 1937)
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(b) deBroglie’s Bohr Model
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Bohr’s model had an ad-hoc
assumption that orbits had quantized
angular momentum (multiples of h-bar)
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deBroglie postulates that only
“standing waves” can yield stationary
orbits, i.e. circumference must be
multiple of the wavelength
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n  2 r
h
nh
p 
Hence allowed momentums are:
 2 r
h
Or angular momentums must
L  rp  n
be quantized:
2
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1c. Phase Velocity
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Velocity of waves are FASTER than light
 E  h 
v  f    
 h  p 
E mc 2 c 2
 

p
mv
v
Where “v” is the
classical speed of the
particle (aka “group
velocity”)
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(d) Interpretation
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deBroglie thought that the “wave” of a
particle had two aspects.
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The “group velocity” described the
localized “particle” nature of the
classical particle
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The “phase velocity” was associated
with the “pilot wave” which traveled
ahead and behind the particle (faster
than light), sensing the environment.
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2. Particle in a Box
a)
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Standing wave patterns
Analogous to waves on
a string with fixed ends.
2L
n 
n
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Momentum hence is
quantized to values:
h
nh
pn  
 2L
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2. Particle in a Box
(b) Energy is hence quantized to
values:
2
2
2
p
nh
En 
 2
2m 8 L m
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The particle can never have zero
energy! The lowest is n=1
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The smaller the box, the bigger the
energy. If wall is height “z”, for
small enough “L”, the particle will
jump and escape!
2
h
 mgz
2
8L m
2c. Wavepackets & Localization
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A wave is infinite in extent, so the “electron” is not localized.
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The superposition of waves of slightly different wavelengths will
create a “localized” wavepacket, which roughly corresponds to
classical particle
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But now it does not have a single momentum (wavelength); it has
a spread of momenta, and the packet will tend to spread out with
time.
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3a. Heisenberg QM
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1925 First formulation of “quantum
mechanics” which correctly describes
energy levels and quantum jumps.
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It’s a mathematical theory, which assumes
that position and momentum do not
commute:
ih
xp  px 
2
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3b. Heisenberg Uncertainty
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“principle of indeterminacy”
“The more precisely the
position is determined, the
less precisely the momentum
is known in this instant, and
vice versa.”
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1927 Uncertainty Principle (which can
be derived from [x,p]=ih …)
h
x p 
4
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C. Wave Mechanics
1. More Quantum Numbers
2. Pauli Exclusion Principle
3. Schrodinger Wave Mechanics
1. Zeeman Effect (1894)
(a) Zeeman effect: splitting of spectral
lines due to magnetic fields, shows us
sunspots have BIG magnetic fields
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1b. Bohr Sommerfeld Model
1916 use elliptical orbits to different energies (new
quantum number “l”).
Also, quantum
number “m” to
describe orientation,
where if l=2, m
could be
{-2,-1,0,1,2}
1c. Bohr’s Periodic Table
1921 uses quantum numbers to explain periodic
table (Pauli’s contribution is that each state has 2
electrons in it, another quantum number)
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2. Pauli Spin
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1924 proposes new quantum number to
explain “Anomalous Zeeman Effect”
where “s” orbits split into 2 lines.
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1925 Uhlenbeck & Goudsmit identify
this as description of “spin” of electron,
which creates a small magnetic moment
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1927 Pauli introduces idea of “spinors”
which describe spin half electrons
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Famous quote: when reviewing a very badly written
paper he criticized it as “It is not even wrong”
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2b. Pauli Exclusion Principle (1925)
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Serious Question: Why don’t all the
electrons fall down into the first (n=1) Bohr
orbit?
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If they did, we would not have the periodic
table of elements!
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Exclusion Principle: Each quantum state
can only have one electron (e.g. 1s orbit
can have two electrons, one with spin up,
other with spin down)
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2c. Fermions & Bosons
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Fermions, which have spin ½ (angular
momentum of h/4) obey the Pauli
exclusion principle (e.g. electrons,
neutrinos, protons, neutrons, quarks)
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Bosons, which have integer spin, do NOT
obey the principle (e.g. photons, gravitons).
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This is why we can have “laser” light (a
bunch of photons with their waves all in
phase).
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3. Schrodinger 1926
Bohr & Heisenberg’s quantum mechanics
used abstract mathematical operations
(e.g. x and p don’t commute)
a) Schrodinger writes a generalized
equation that deBroglie waves must
obey when there is Potential Energy
(such that the wavelength changes from
point to point in space)
h 


V
(
x
)


E

2
2m x
2
2
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3b Electron Orbits
• S orbits hold 2 electrons
• P orbits hold 6 electrons
• D orbits hold 10 electrons
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Electron Configurations
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Bohr’s Aufbau (build up) Principle: Fill
orbits of lowest energy first (e.g. the n=1
orbit before the n=2 orbit)
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Madelung Rule: for states (n,l), the
states with lower sum “n+l” are filled
first (because they have lower energy).
For example, 4s (4,0) would be filled before
3d (3,2).
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Hund’s Rules (Bohr’s assistant)
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Madelung Rule
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Hund’s Rules
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1. Rule of Maximum Multiplicity: maximize the
spin (e.g. put one electron into each of the three
p orbits with spins parallel, i.e. maximize
unpaired electrons).
2. For a given multiplicity, the term with the
largest value of L (orbital angular
momentum), has the lowest energy
3. The level with lowest energy (where J=L+S)
1. Outer shell Less than half filled: minimum J
2. Outer shell more than half filled: maximum J
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3c. Max Born
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1924 coins the term “Quantum Mechanics”
1925 helps with Heisenberg’s matrix form
of quantum mechanics
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1928 The square of the quantum wave
is proportional to the probability of
finding the particle at that position.
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Hence you can think of the quantum
wave as having a “classical” probability
density , and an “imaginary” quantum
phase part.
  e
i
References/Notes
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McEvoy & Zarate, “Introducing Quantum Theory” (Totem Books, 1996)
http://www.aip.org/history/heisenberg/p08.htm (includes audio !)
http://www.uky.edu/~holler/html/orbitals_2.html
http://www.meta-synthesis.com/webbook/30_timeline/lewis_theory.php
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