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
Massac husetts Institute of T ec hnology
Par t II T opics
1. It ’ s a Quantum W orld: The Theor y of Quantum Mechanics
2. Quantum Mechanics: Practice Mak es P erf ect
3. Fr om Man y-Body to Single-Par ticle; Quantum Modeling of Molecules
4. Application of Quantum Modeling of Molecules: Solar Thermal Fuels
5. Application of Quantum Modeling of Molecules: Hydr ogen Storage
6. Fr om Atoms to Solids
7. Quantum Modeling of Solids: Basic Pr oper ties
8. Advanced Pr op . of Materials: What else can w e do?
9. Application of Quantum Modeling of Solids: Solar Cells Par t I
10. Application of Quantum Modeling of Solids: Solar Cells Par t II
11. Application of Quantum Modeling of Solids: Nanotechnolog y
Motivation
electr on in bo x
?
Image adapted from Wikimedia Commons , http://commons.wikimedia.org .
Image of NGC 604 nebula is in the public domain. Source: Hubble Space Telescope Institute (NASA). Via Wikimedia Commons .
Lesson outline
• Re vie w
• A r eal w orld example
• Ev er ything is spinning
• Pauli ’ s exclusion
T his image is in the public domain. Source: Wikimedia Commons .
• P eriodic table of elements
R e vi e w: W h y QM?
P r oblems in classical p h ysics that led to quantum mechanics:
• “classical atom”
• quantization of p r oper ties
• w a v e aspect of matter
• (black-body radiation), ...
R e vi e w: Quantization
_
_
_ _
_
_
_
_
_
_
_
_
_
photoelectric E e
eff ect
r A
Image by MIT OpenCourseWare.
E = n ( ⇥ - ⇥ A ) = h ( v - v A )
h = 2 7 n = 6 . 6 · 10 - 34 W atts ec. 2
Einstein: photon E = n w
“Classical atoms”
e -
+
p r oblem:
accelerated charge causes radiation, atom not stable!
h yd r ogen atom
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R e vi e w: W a v e aspect
light matter
w a v e character par ticle character
_
_
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_
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_
_
_
_
_
Image by MIT OpenCourseWare.
Image in public domain. See Wikimedia Commons .
Double-Slit
Courtesy of Bernd Thaller. Used with permission.
R e vi e w: W a v e aspect
par ticle : E and momentum p ⇥
w a v e: fr equency and w a v e v ector ⇥ k
E = h v = n ⇥
p ⇧ = n ⇧ k =
h ⇧ k
A | ⇧ k |
de B r oglie: fr ee par ticle can be described a as
plane w a v e
1 ( ⌥ r , t ) = Ae i ( ⌃ k · ⌃ r - w t )
with
A = h mv
11
R e vi e w: Interp r etation of QM
1 ( ⇧ r , t ) w a v e function (complex)
| 1 | 2 = 1 1 * interpr etation as pr obability to find par ticle!
(r, t)
2
(r, t)
∫
⇤
1 1 ⇥ dV = 1
— ⇤
Image by MIT OpenCourseWare.
W a v e Par ticles Hitting a W all
Courtesy of Bernd Thaller. Used with permission.
R e vi e w: Schrödinger equation
a w a v e equation:
second derivativ e in space
first derivativ e in time
k 2
— 2 m D
2
+ V ( r , t ) G ( r , t ) = i k G ( r , t )
6 t
⇥
6→
2
1 2
H = - 2 m ⇥
+ V ( r , t ) =
Hamiltonian
p 2 p ⇤ = - i 1 ⇥
= 2 m
+ V = T + V
Schrödinge r ...
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16
R e vi e w: Schrödinger equation
H time independent: 1 ( ⌃ r , t ) = 1 ( ⌃ r ) · f ( t )
f ˙ ( t ) H 1 ( r r )
i n = = const. = E f ( t ) 1 ( r r )
H G ( → r ) = E G ( → r )
k
/ ( ⌃ r , t ) = / ( ⌃ r ) · e -
i E t
time independent Schrödinger equation stationar y Schrödinger equation
Par ticle in a b o x
Schrödinger equation
boundar y conditions general solution
It ’ s r eal!
(theor y)
Ti-O Bond
Cu-O Bond (experiment)
Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Zuo, J., M. Kim, et al. "Direct O bservation of d-orbital
19 Holes and Cu-Cu Bonding in Cu2O." Nature 401, no . 6748 (1999): 49-52. © 1999.
Screenshot of Scientific American article "Observing Orbitals" removed due to copyright restrictions; read the article online .
What ’ s this g ood f or?
Image in the public domain.
Hyd r ogen: a r eal w orld example .
The Hyd r ogen Futu r e?
Image s in the public domain.
Histor y of Hyd r ogen
© ACS Publications. From: Grochala, W., and Peter P. Edwards. Chemical Reviews 104
23 (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 / .
Ho w large of a gas tank do w e want?
The “D r op T est”
.
Figure 1 © Toyota Motor Corporation, "Drop Test" © EDO Canada. All rights reserved. This content is excluded
The h yd r ogen atom
?
elect r ostatics:
Coulomb potential
r
+
e -
stationar y Schrödinger equation
w a v e functions possible energies
The h yd r ogen atom
stationar y
Schrödinger equation H l = E l
T + V ⇥ G = E G
+ V ⇥ G ( → r ) = E G ( → r )
0
e 2 ⇥
— 2 m D
— 4 v s r
G ( → r ) = E G ( → r )
The h yd r ogen atom
choose a mor e suitable coor dinate system:
spherical coor dinates
G ( → r ) = G ( x, y , z )
= G ( r , θ , ф )
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The h yd r ogen atom
Schrödinger equation in spherical coor dinates:
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28
The h yd r ogen atom
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solv e b y separation of variables:
The h yd r ogen atom
separation of variables
The h yd r ogen atom
R ( r )
Solution exists
if and only i f .....
n = 1, 2, 3 .........
Main quantum number
P ( )
Solution exists
if and only i f .....
l = 0, 1, 2, 3 n -1
Orbital qua ntum number
F ( )
Solution exists
if and only i f .....
m l = - l , - l +1,... + l
Magnetic qua ntum number
Image by MIT OpenCourseWare.
The h yd r ogen atom
q u a n t u m n u m b e r s
|
n |
l |
m l |
F( ) |
P( ) |
R(r) |
|
1 |
0 |
0 |
1 2 |
1 2 |
2 e -r/a 0 a 3/ 2 0 |
|
2 |
0 |
0 |
1 2 |
1 2 |
1 r -r / 2 a 0 2 2 a 3/ 2 2- a e 0 0 |
|
2 |
1 |
0 |
1 2 |
6 co s 2 |
1 r e -r / 2 a 0 2 6 a 3/ 2 a 0 0 |
|
2 |
1 |
1 |
1 e ±i 2 |
3 sin 2 |
1 r e -r / 2 a 0 2 6 a 3/ 2 a 0 0 |
Image by MIT OpenCourseWare.
The h yd r ogen atom
standar d notation f or states:
|
"Sharp" |
s |
l = 0 |
|
"Principal" |
p |
l = 1 |
|
"Diff use" |
d |
l = 2 |
|
"Funda mental" |
f |
l = 3 |
For example, if n = 2, l = 1, the state is designated 2p
Image by MIT OpenCourseWare.
The h yd r ogen atom
q u a n t u m n u m b e r s
|
n |
l |
m l |
F( ) |
P( ) |
R(r) |
|
1 |
0 |
0 |
1 2 |
1 2 |
2 e -r/a 0 a 3/ 2 0 |
|
2 |
0 |
0 |
1 2 |
1 2 |
1 r -r / 2 a 0 2 2 a 3/ 2 2- a e 0 0 |
|
2 |
1 |
0 |
1 2 |
6 co s 2 |
1 r e -r / 2 a 0 2 6 a 3/ 2 a 0 0 |
|
2 |
1 |
1 |
1 e ±i 2 |
3 sin 2 |
1 r e -r / 2 a 0 2 6 a 3/ 2 a 0 0 |
Image by MIT OpenCourseWare.
The h yd r ogen atom
http://ww w .orbitals.com/orb/orbtable .htm
Courtesy of David Manthey. Used with permission. Source: http://www.orbitals.com/orb/orbtable.htm .
The h yd r ogen atom
Energies:
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The h yd r ogen atom
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38
The h yd r ogen atom
39 © R. Nave. All rights reserved. This content is excluded from our Creative Commons
Atomic units
1 eV = 1.6021765 -19 J
1 Rydberg = 13.605692 eV = 2.1798719 -18 J
1 Har tr ee = 2 Rydberg
1 Bohr =5.2917721 -11 m
Atomic units (a.u.) :
Energies in Ry Distances in Bohr
Also in use: 1 Å =10 -10 m, nm= 10 -9 m
40
Slightl y Inc r eased Complexity
H G ( → r ) = E G ( → r )
Anal ytic solutions become extr emel y complicated, e v en f or simple systems.
41
Next? Helium!
e -
r 1 r 12
-
H 1 = E 1
+ e
r 2 H 1
+ H 2
+ W ⇥ G ( → r 1
, → r 2 ) = E G ( → r 1
, → r 2 )
T 1 + V 1 + T 2 + V 2 + W ⇥ G ( → r 1 , → r 2 ) = E G ( → r 1 , → r 2 )
k 2 2
e 2 k 2 2 e 2
e 2 ⇥
0
— 2 m D 1 —
4 v s 0 r 1
— 2 m D 2 —
4 v s 0 r 2
+ 4 v s
r 12
G ( r 1 , r 2 ) = E G ( r 1 , r 2 )
cannot be solv ed anal yticall y
p r oblem!
Solution in general?
Onl y a f e w p r oblems ar e solvable anal yticall y .
W e need a pp r o ximate a pp r oaches:
per turbation theor y
matrix eigen value equation
Solution in general?
P er turbation theor y:
small
H = H 0 + Z H 1
w a v e functions and energies ar e kno wn
w a v e functions and energies will be similar to those of H o
Solution in general?
Matrix eigen value equation:
H 1 = E 1
H Σ c i c i = E Σ c i c i
⇥ = Σ c i c i
i
expansion in or thonormalized basis functions
i
∫
∫ d ⌥ r c H Σ Σ
i
c i c i = E d ⌥ r c c i c i
j j
i i
Σ H j i c i = E c j
i
H ⇤ c = E ⇤ c
E v er ything is spinning ...
Stern–Gerlach experiment (1922)
F ⇧ = — D E
= D m ⇧ · B ⇧
Image courtesy of Teresa Knott.
E v er ything is spinning ...
In quantum mechanics par ticles can h a v e a magnetic moment and a ”spin”
magnetic moment
m ⇥
spinning charge
E v er ything is spinning ...
conclusion f r om the Stern-Gerlach experiment
f or elect r ons: spin can ON L Y be
up d o wn
E v er ything is spinning ...
ne w quantum n umber : spin quantum n umber f or elect r ons: spin quantum n umber can ON L Y be
up d o wn
Spin Histor y
Disco v er ed in 1926 b y Goudsmit and Uhlenbeck
Part of a letter by L.H. Thomas to Goudsmit on Mar ch 25 1926 (sour ce: W ikipedia).
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50
Pauli ’ s exclusions principle
T w o elect r ons in a system cannot h a v e the same quantum n umbers!
quantum n umbers: main n: 1,2,3 ...
orbital l: 0,1,...,n-1
magnetic m: -l,...,l spin: up , do wn
h ydr ogen
... ... ... ...
3s 3p 3d
2s 2p
1s
P eriodic table of elements
Connection to materials?
optical p r oper ties of gases
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R e vi e w
• Re vie w
• A r eal w orld example!
• Ev er ything is spinning
• Pauli ’ s exclusion
• P eriodic table of elements
This image is in the public domain. Source: Wikimedia Commons .
Literatu r e
• Gr eine r , Quantum Mechanics: An Int r oduction
• F e ynman, The F e ynman Lectur es on P h ysics
• wikipedia, “ h yd r ogen atom”, “Pauli exclusion principle”,
“periodic table”, ...
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3 . 021 J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modelling and Simulation
Spring 20 1 2
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