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Transcript
1
Quantum Mechanics:
Wave Theory of Particles
Dr. Bill Pezzaglia
QM Part 2
Updated: 2012Aug28
2
Quantum Mechanics
A. Bohr Model of Atom
B. Wave Nature of Particles
C. Schrodinger Wave Equation
3
A. Bohr Model of Atom
1. Bohr’s First Postulate
•
•
•
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
4
1. Bohr’s First Postulate
(a) Quantized Angular Momentum
•
•
1912 first ideas by J.W. Nicholson
Postulates angular momentum of electron in
atom must be a multiple of
h

L  mvr  n
2
5
1. Bohr’s First Postulate
(b) Stationary Orbits
•
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
•
Principle quantum number “n” has a lowest
value of n=1 (lowest angular momentum of one
h-bar).
(c) The Bohr Radius
With only a little algebra can solve for radius of
electron’s orbit in the atom. Details (can ignore!):
•
Classical equation of motion
•
Substitute:
•
Solve for radius:
•
Bohr Radius:
v2
( Ze)e
m 
r 40 r 2
L
n
v

mr mr
a0
rn  n
Z
2
h 2 0
a0 
 0.053 nm
2
me
6
7
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
8
(b) Energy of nth orbit
DETAILS (ignore)
• Viral Theorem: For
inverse square law force:
•
•
Hence total energy:
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
DETAILS (ignore)
•
Substitute Bohr’s radius formula for n-th orbit
gives energy of nth orbit:
Z 2 hcR
Z2
En  
 (13.6 ev) 2
2
n
n
•
Importance is Bohr was able to calculate
Rydberg’s constant from scratch!
e2
me4
R
 3 2
80 hca0 8h  0 c
9
(c) Bohr Derives Balmer’s Formula
•
From Einstein-Planck Formula:
hf 
•
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
10
11
3. Bohr’s Correspondence Principle
•
1923: Classical mechanics “corresponds” to
quantum system for BIG quantum numbers.
•
When “n” is big, it behaves classically
•
When “n” is small, it behaves “quantumly” (is
that a word?)
12
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

•
Proof: 1927 Electron diffraction
experiment of Davisson & Germer
(Nobel Prize 1937)
13
(b) deBroglie’s Bohr Model
DETAILS (ignore)
•
Bohr’s model had an ad-hoc
assumption that orbits had quantized
angular momentum (multiples of h-bar)
•
•
•
deBroglie postulates that only
“standing waves” can yield stationary
orbits, i.e. circumference must be
multiple of the wavelength
n  2 r
h
nh
p 
 2 r
Hence allowed momentums are:
h
L  rp  n
Or angular momentums must
2
be quantized:
14
2. Particle in a Box
a)
•
Standing wave patterns
Analogous to waves on
a string with fixed ends.
2L
n 
n
•
Momentum hence is
quantized to values:
h
nh
pn  
 2L
15
16
2. Particle in a Box
(b) Energy is hence quantized to
values:
p 2 n2h2
En 
 2
2m 8 L m
•
The particle can never have zero
energy! The lowest is n=1
•
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
•
A wave is infinite in extent, so the “electron” is not localized.
•
The superposition of waves of slightly different wavelengths will
create a “localized” wavepacket, which roughly corresponds to
classical particle
•
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.
17
3. Heisenberg Uncertainty
•
•
“principle of indeterminacy”
“The more precisely the
position is determined, the
less precisely the momentum
is known in this instant, and
vice versa.”
•
1927 Uncertainty Principle (which can
be derived from [x,p]=ih …)
h
x p 
4
18
19
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
20
1b. Angular Momentum
Quantum Number
Zeeman effect implies “suborbits” which are
affected differently by the magnetic field.
• Principle (Bohr) quantum number n=1, 2, 3, 4 …
• 2nd quantum number “l” where l <n
n=1
n=2
n=3
l =0 (“s” orbit)
l =0 or l=1 (“p” orbit)
l =0, 1, or 2 (“d” orbit)
21
22
Multiplicity of states
3rd quantum number “m”
l =0 (“s” orbit)
l =1 (“p” orbit)
l =2 (“d” orbit)
m=0
m=-1, 0, +1
m=-2, -1, 0, +1, +2
So the Zeeman effect is splitting the “p” orbits into
three different lines (and “d” orbits into 5)
2. Pauli Spin
•
1924 proposes new quantum number to
explain “Anomalous Zeeman Effect”
where “s” orbits split into 2 lines.
•
1925 Uhlenbeck & Goudsmit identify
this as description of “spin” of electron,
which creates a small magnetic moment
•
1927 Pauli introduces idea of “spinors”
which describe spin half electrons
•
Famous quote: when reviewing a very badly written
paper he criticized it as “It is not even wrong”
23
2b. Pauli Exclusion Principle (1925)
•
Serious Question: Why don’t all the
electrons fall down into the first (n=1) Bohr
orbit?
•
If they did, we would not have the periodic
table of elements!
•
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)
24
2c. Fermions & Bosons
Details
• Fermions, which have spin ½ (angular
momentum of h/4) obey the Pauli exclusion
principle (e.g. electrons, neutrinos, protons,
neutrons, quarks)
•
Bosons, which have integer spin, do NOT obey
the principle (e.g. photons, gravitons).
•
This is why we can have “laser” light (a bunch of
photons with their waves all in phase).
25
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
26
3b Solution to Schrodinger Equation:
Electron Orbits
• S orbits hold 2 electrons
• P orbits hold 6 electrons
• D orbits hold 10 electrons
27
Madelung Rule: Filling electrons into an atom
28
Electron Configurations
•
Bohr’s Aufbau (build up) Principle: Fill orbits of lowest
energy first (e.g. the n=1 orbit before the n=2 orbit)
•
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).
•
•
Carbon: 1s2, 2s2, 2p2
Titanium: 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3p2
29
References/Notes
•
•
•
•
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
30