1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012
Part II – Quantum Mechanical Methods : Lecture 2
Quantum Mechanics:
Practice Makes Perfect
Jeffrey C. Grossman
Department of Materials Science and Engineering
Massachusetts Institute of Technology
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Part II Topics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
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It’s a Quantum World:The Theory of Quantum Mechanics
Quantum Mechanics: Practice Makes Perfect
From Many-Body to Single-Particle; Quantum Modeling of Molecules
Application of Quantum Modeling of Molecules: Solar Thermal Fuels
Application of Quantum Modeling of Molecules: Hydrogen Storage
From Atoms to Solids
Quantum Modeling of Solids: Basic Properties
Advanced Prop. of Materials:What else can we do?
Application of Quantum Modeling of Solids: Solar Cells Part I
Application of Quantum Modeling of Solids: Solar Cells Part II
Application of Quantum Modeling of Solids: Nanotechnology
Motivation
electron in box
?
Image of NGC 604 nebula is in the public domain. Source: Hubble
Space Telescope Institute (NASA). Via Wikimedia Commons.
Image adapted from Wikimedia Commons, http://commons.wikimedia.org.
3
Lesson outline
• Review
• A real world example
• Everything is spinning
• Pauli’s exclusion
• Periodic table of elements
This image is in the public domain. Source: Wikimedia Commons.
4
Review:Why QM?
Problems in classical physics that
led to quantum mechanics:
• “classical atom”
• quantization of properties
• wave aspect of matter
• (black-body radiation), ...
5
Review: Quantization
photoelectric
effect
Ee
_
_
_
_
_
_
_
_
_
_
_
_
_
Image by MIT OpenCourseWare.
E = n(⇥ - ⇥A ) = h(v - vA )
A
h = 27n = 6.6 · 10-34 Wattsec.2
Einstein: photon E = nw
6
w
“Classical atoms”
e+
hydrogen atom
7
problem:
accelerated charge causes
radiation, atom not stable!
From Wikipedia. License: CC-BY-SA. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Review:Wave aspect
light
matter
wave character
particle character
_
_
_
_
_
_
_
_
_
_
_
_
_
Image by MIT OpenCourseWare.
Image in public domain. See Wikimedia Commons.
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Double-Slit
Courtesy of Bernd Thaller. Used with permission.
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Review:Wave aspect
particle:E and momentum p⇥
wave: frequency v and wavevector ⇥k
E = hv = n⇥
⇧
h
k
k=
p
⇧ = n⇧
A |⇧
k|
de Broglie: free particle can be described a as
h
i(⌃
k·⌃
r -wt)
planewave 1(⌥r, t) = Ae
with A =
mv
11
s
Review: Interpretation of QM
hoscom2.gif 512!342 pixels
1(⇧
r , t)
|1| = 11
2
3/24/
wave function (complex)
*
interpretation as probability to find particle!
⇤
�(r, t)
�(r, t)
2
Image by MIT OpenCourseWare.
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⇤
11 ⇥ dV = 1
Wave Particles Hitting
a Wall
Courtesy of Bernd Thaller. Used with permission.
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Electron Wave/Particle Video
Courtesy of cassiopeiaproject.com.
14
Source:YouTube (cassiopeiaproject)
Review: Schrödinger
equation
second derivative in space
first derivative in time
a wave equation:
2
2m
Ha
H
mi
15
lto
nia
n
⇥ + V (r, t)
2
12
= =
2m
2
p
2m
⇥
(r, t) = i
⇥ + V (r, t) =
⇤
⇤t
(r, t)
2
+V =T +V
p
⇤ = -i1⇥
Schrödinger...
From Wikipedia. License: CC-BY-SA. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
16
Review: Schrödinger
equation
H time independent:
in
f˙(t)
f (t)
=
1(⌃
r , t) = 1(⌃
r ) · f (t)
H1(r
r)
1(r
r)
H (⇧
r ) = E (⇧
r)
time independent Schrödinger equation
stationary Schrödinger equation
17
= const. = E
/(⌃
r , t) = /(⌃
r) · e
- i Et
Particle in a box
Schrödinger equation
boundary conditions
18
general solution
It’s real!
Cu-O Bond
(experiment)
19
Reprinted by permission from Macmillan Publishers Ltd: Nature.
Source: Zuo, J., M. Kim, et al. "Direct Observation of d-orbital
Holes and Cu-Cu Bonding in Cu2O." Nature 401, no. 6748
(1999): 49-52. © 1999.
Ti-O Bond
(theory)
Screenshot of Scientific American article "Observing Orbitals" removed due to copyright restrictions; read the article online.
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What’s this good for?
Hydrogen:
a real world
example.
Image in the public domain.
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The Hydrogen Future?
Images in the public domain.
22
History of
Hydrogen
23
© ACS Publications. From: Grochala, W., and Peter P. Edwards. Chemical Reviews 104
(2004): 1283-1315. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
How large of a gas tank
do we want?
The “Drop Test”
.
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Figure 1 © Toyota Motor Corporation, "Drop Test" © EDO Canada. All rights reserved. This content is excluded
from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
The hydrogen atom
er
+
electrostatics:
Coulomb potential
?
stationary
Schrödinger equation
wave functions
possible energies
25
The hydrogen atom
stationary
Schrödinger equation
Hl = El
T +V
2
e
v
l
o
ts
jus
2m
☺
2
2m
26
⇥
2
⇥2 + V
e2
4⇥
0r
⇥
⇥
⇥
=E
(⌃
r ) = E (⌃
r)
⇤(r) = E⇤(r)
The hydrogen atom
choose a more suitable
coordinate system:
spherical coordinates
(⌥
r) =
(x, y, z)
= ⇤(r, , ⇥)
© R. Nave. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
27
The hydrogen atom
Schrödinger equation in spherical coordinates:
© R. Nave. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
28
The hydrogen atom
solve by separation
of variables:
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license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
29
The hydrogen atom
separation
of variables
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The hydrogen atom
R(r)
Solution exists
if and only if.....
n = 1, 2, 3 .........
P(Ψ)
Solution exists
if and only if.....
l = 0, 1, 2, 3 ....n-1
F(�)
Solution exists
if and only if.....
ml = -l, -l+1,... +l
Main quantum number
Orbital quantum number
Magnetic quantum number
Image by MIT OpenCourseWare.
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The hydrogen atom
quantum numbers
n
1
2
2
2
l
0
0
1
1
ml
F(φ)
P(θ)
0
1
2π
1
2
a03/2
0
1
2π
1
2

 -r / 2a
1
r
0
2e
2 2a03/2  a0 
0
1
2π
6
cos θ
2
1
r -r / 2a0
e
3/2 a
2 6 a0
0
1 ±iφ
e
2π
3
sin θ
2
1
r -r / 2a0
e
3/2 a
2 6 a0
0
±1
R(r)
2
e -r/a0
Image by MIT OpenCourseWare.
32
The hydrogen atom
standard notation for states:
"Sharp"
s
"Principal"
p
l = 0 For example, if n = 2, l = 1,
the state is designated 2p
l = 1
"Diffuse"
d
l = 2
"Fundamental" f
l = 3
Image by MIT OpenCourseWare.
33
The hydrogen atom
quantum numbers
n
1
2
2
2
l
0
0
1
1
ml
F(φ)
P(θ)
0
1
2π
1
2
a03/2
0
1
2π
1
2

 -r / 2a
1
r
0
2e
2 2a03/2  a0 
0
1
2π
6
cos θ
2
1
r -r / 2a0
e
3/2 a
2 6 a0
0
1 ±iφ
e
2π
3
sin θ
2
1
r -r / 2a0
e
3/2 a
2 6 a0
0
±1
R(r)
2
e -r/a0
Image by MIT OpenCourseWare.
34
The hydrogen atom
http://www.orbitals.com/orb/orbtable.htm
Courtesy of David Manthey. Used with permission. Source: http://www.orbitals.com/orb/orbtable.htm.
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36
Courtesy of David Manthey. Used with permission. Source: http://www.orbitals.com/orb/orbtable.htm.
The hydrogen atom
Energies:
37
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license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
The hydrogen atom
38
© unknown. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
The hydrogen atom
39
© R. Nave. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Atomic units
1 eV =
-19
1.6021765
J
1 Rydberg = 13.605692 eV = 2.1798719-18 J
1 Hartree = 2 Rydberg
1 Bohr =5.2917721-11 m
Atomic units (a.u.):
Also in use:
40
Energies in Ry
Distances in Bohr
-10
1Å=10 m, nm=
-9
10 m
Slightly Increased Complexity
H (⇧
r ) = E (⇧
r)
Analytic solutions become
extremely complicated, even
for simple systems.
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Next? Helium!
er1
+
r2
r12
e-
H1 = E1
H 1 + H2 + W
T1 + V1 + T2 + V2 + W
2
2m
⇥21
e2
4⇥
0 r1
2
2m
⇥22
cannot be solved analytically
42
e2
4⇥
0 r2
+
⇥
⇥
(⌥
r1 , ⌥
r2 ) = E (⌥
r1 , ⌥
r2 )
(r1 , r2 ) = E (r1 , r2 )
e2
4⇥
0 r12
⇥
⇤(r1 , r2 ) = E⇤(r1 , r2 )
problem!
Solution in general?
Only a few problems are solvable analytically.
We need approximate approaches:
perturbation theory
43
matrix eigenvalue
equation
Solution in general?
Perturbation theory:
small
H = H0 + H1
wave functions and
energies are known
wave functions and energies
will be similar to those of Ho
44
Solution in general?
⇥=
Matrix eigenvalue equation:
i
expansion in
orthonormalized basis
functions
H1 = E1
ci ci = E
H
⇥
i
ci ci = E
d⌥
r cj H
i
ci ci
⇥i
Hji ci = Ecj
i
H⇤
c = E⇤
c
45
ci ci
ci ci
d⌥
r cj
i
Everything is spinning ...
Stern–Gerlach experiment (1922)
⇧
F
=
⇤E
⇧
= ⇤m
⇧ ·B
Image courtesy of Teresa Knott.
46
Everything is spinning ...
In quantum mechanics particles can have
a magnetic moment and a ”spin”
magnetic
moment
47
m
⇥
spinning
charge
Everything is spinning ...
conclusion from the
Stern-Gerlach experiment
for electrons: spin can ONLY be
up
48
down
Everything is spinning ...
new quantum number: spin quantum number
for electrons: spin quantum number can ONLY be
up
49
down
Spin History
Discovered in 1926 by
Goudsmit and Uhlenbeck
Part of a letter by L.H. Thomas to Goudsmit on March 25 1926 (source: Wikipedia).
© Niels Bohr Library & Archives, American Institute of Physics. All rights reserved. This content is excluded
from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
50
Pauli’s exclusions
principle
Two electrons in a system cannot have
the same quantum numbers!
hydrogen
quantum numbers:
main n: 1,2,3 ...
orbital l: 0,1,...,n-1
magnetic m: -l,...,l
spin: up, down
51
...
3s
2s
1s
... ...
3p 3d
2p
...
Periodic table of
elements
52
This image is in the public domain. Source: Wikimedia Commons.
Connection to
materials?
optical properties of gases
© unknown. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
53
Review
• Review
• A real world example!
• Everything is spinning
• Pauli’s exclusion
• Periodic table of elements
54
This image is in the public domain. Source: Wikimedia Commons.
Literature
• Greiner, Quantum Mechanics:
An Introduction
• Feynman,The Feynman Lectures
on Physics
• wikipedia,“hydrogen atom”,
“Pauli exclusion principle”,
“periodic table”, ...
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MIT OpenCourseWare
http://ocw.mit.edu
3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modelling and Simulation
Spring 2012
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