Quantum mechanics review
• Reading for week of 1/28-2/1
– Chapters 1, 2, and 3.1,3.2
• Reading for week of 2/4-2/8
– Chapter 4
Schrodinger Equation (Time-independent)


H n  E n
where



 T V
H

 
H
2
2m

 V
2
The solutions incorporate boundary conditions and are a family of
eigenvalues with increasing energy and corresponding
eigenvectors with an increasing number of nodes.
The solutions are orthonormal.
 
*
n
m
d   nm
Physical properties: Expectation values

A    A n d
*
n
or

A  n An
Dirac notation or bra-ket notation
Physical properties: Hermitian Operators
Real Physical Properties are Associated with Hermitian Operators
Hermitian operators obey the following:

A
mn
 m An  A

nm
 n Am
The value <A>mn is also known as a matrix element, associated with
solving the problem of the expectation value for A as the eigenvalues
of a matrix indexed by m and n
Zero order models:
Particle-in-a-box: atoms, bonds,
conjugated alkenes, nano-particles
Harmonic oscillator: vibrations of atoms
Rigid-Rotor: molecular rotation; internal
rotation of methyl groups, motion within
van der waals molecules
Hydrogen atom: electronic structure
Hydrogenic Radial Wavefunctions
Particle-in-a-3d-Box
V(x) =0; 0<x<a
V(x) =∞; x>a; x <0
V(x)
b y ; c  z
x
a



H  T V
2
2
2







 V
 
 V   


2m
2m  x 2 y 2 z 2 


nx n y nz
2
2

2
8
 nxx   n yy   nzz 
 sin

sin
 sin

abc  a   b   c 
nx,y,z = 1,2,3, ...
Particle-in-a-3d-Box
V(x) =0; 0<x<a
V(x) =∞; x>a; x <0
V(x)
b y ; c  z
x
a
h 2  nx
 2

8m  a

2
Enx n y nz
2
  n y   nz 2  
   2    2 
  b   c 

 
 
2
2
2 







1
1x
h
1
y 
z

 2   2   2   0
E111 
8m  a   b   c 


2
Zero point energy/Uncertainty Principle
In this case since V=0 inside the box E = K.E.
If E = 0 the p = 0 , which would violate the uncertainty principle.

xp 
2
Zero point energy/Uncertainty Principle
More generally
Variance or rms:
 2

A
A

If the system is an eigenfunction of
there is no variance.
A
     
 x, p   x p  p x  i


2
A

then
A is precisely determined and

xp 
2
Zero point energy/Uncertainty Principle
1   
A B 
A
,
B

2 
If the commutator is non-zero then the two properties cannot be
precisely defined simultaneously. If it is zero they can be.
Harmonic Oscillator 1-d
F=-k(x-x0)
Internal coordinates; Set x0=0


d
H  
V ;
2
2 dx
2
2

2
1
V
kx
2
Harmonic Oscillator Wavefunctions
 v  N v H v   x e
0.5x 2
Hermite polynomials
H v q   (1) v e q
2
n
d q 2
e
n
dq
H 0 q   1
H1 q   2q
H 2 q   2  4q 2
H 3 q   12q  8q 3
H 4 q   12  48q 2  16q 4
H 5 q   120q  16q 3  32q 5

N v    v
2 v!  

1
2
V = quantum number = 0,1,2,3
   / 
Hv = Hermite polynomials
Nv = Normalization Constant
 0  N0e

0.5x 2

1  N1 2  x e
0.5x 2
Ev  (v  12)  
k

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html#c1
Raising and lowering operators:
Recursion relations used to define new members in a family of solutions to D.E.
 0   ipˆ 
a  
 X 
 raising
2

0 


 0   ipˆ 
a  
X
lowering



2 
0 

 V  1 V  1
  V V 1
a V 
a V
Rotation: Rigid Rotor


2
L
H
2I
I  R

2
2

2
1  
L   r p 




2

2
L  L x L y L
 2  
L , Li   0


z
V 0
Rotation: Rigid Rotor
Wavefunctions are the spherical harmonics
Ylm  ,   
(1)
m m
2l  1
4
lm  Ylm , 
l  m !P
l  m !
l
m
(cos )eim
Operators L2 ansd Lz

L z Ylm  m Ylm

2
L Ylm  l (l  1) Ylm
Degeneracy
Angular Momemtum operators the spherical harmonics
Operators L2 ansd Lz

L z Ylm  m Ylm

2
L Ylm  l (l  1) Ylm

l ' m' L z lm  m m 'm l 'l


2
l ' m' L lm  l (l  1) m 'm l 'l
Rotation: Rigid Rotor


2
L
H
2I
I  R
2
1  
L   r p 



V 0
Eigenvalues are thus:

2
L
l (l  1)
l ' m'
lm 
 m 'm l 'l
2I
2I
l = 0,1,2,3,…
Lots of quantum mechanical and spectroscopic problems
have solutions that can be usefully expressed as sums of
spherical harmonics.
e.g.
coupling of two or more angular momentum
plane waves
reciprocal distance between two points in space
Also many operators can be expressed as spherical
harmonics:
l ' m' YLM lm
The properties of the matrix element above are well known and are zero unless
-m’+M+m = 0
l’+L+l is even
Can define raising and lowering operators for these wavefunctions too.
The hydrogen atom

H
2
e
 
 
2m
r
2
2
Set up problem in spherical polar coordinates. Hamiltonian is
separable into radial and angular components
nlm  Rnl r Ylm ,  
n
the principal quantum number,
determines energy
l
the orbital angular momentum
quantum number
l= n-1, n-2, …,0
m
the magnetic quantum number
-l, -l+1, …, +l
e
4
R
En   2 2   2 ; R  13.6eV  1312kJ / m ole
2 n
n