Physics 451
Quantum mechanics I
Fall 2012
Oct 12, 2012
Karine Chesnel
Announcements
Quantum mechanics
Announcements
Homework next week:
• HW # 13 due Tuesday Oct 16
Pb 3.3, 3.5, A18, A19, A23, A25
• HW #14 due Thursday Oct 18
Pb 3.7, 3.9, 3.10, 3.11, A26
Quantum mechanics
Hilbert space
Infinite- dimensional space
N-dimensional space

e1 , e2 , e3 ,... eN


1
,  2 ,  3 ...  n ...

Wave function are normalized:
  ( x)
2
dx  1

b
Hilbert space: functions f(x) such as

f ( x) dx  
2
a
Wave functions live in Hilbert space
Quantum mechanics
Hilbert space

f g 
Inner product

f * ( x) g ( x) dx


Norm
f
2
 f f 

f * ( x ) f ( x )dx

f m f n   nm
Orthonormality
f g  f
Schwarz inequality



g

f ( x)* g ( x)dx 



f ( x) dx  g ( x) dx
2
2

Quantum mechanics
Hermitian operators
Observable - operator

Q 
Expectation value

 *Qdx   Q 

since
Q  Q
*
 Q    Q  Q 
For any f and g functions
f Qg  Qf g 
Q†  Q
Observables are Hermitian operators
Examples:
x̂
p̂
Quantum mechanics
Determinate states
Stationary states – determinate energy
H  n  En  n
Generalization of
Determinate state:
Standard deviation:
Q q
 2  Q2  Q
 2   Q  Q
For determinate state:

2
2

Q  Q
   Q  Q
Q 
 0
Q

Q  Q 
operator
eigenvalue
eigenstate

 Q 
†
0
2
Q

Quantum mechanics
Quiz 16
Since any wave function can be written as a linear combination
of determinate states (stationary states), for which we can write
H  n  En  n
The wave function is itself a determinate state and we can write
H  H 
A. True
B. False
Quantum mechanics
Eigenvectors & eigenvalues
For a given transformation T, there are “special” vectors for which:
T a  a
a
is transformed into a scalar multiple of itself
a
is an eigenvector of T
 is an eigenvalue of T
Quantum mechanics
Eigenvectors & eigenvalues
To find the eigenvalues:
T   I  a
 0
det T   I   0
We get a Nth polynomial in : characteristic equation
Find the N roots
 1, 2 ,...N 
Spectrum
Quantum mechanics
Hermitian transformations
 T    T   T † 
Hermitian operator:
T T
†
1. The eigenvalues are real
2. The eigenvectors corresponding to distinct eigenvalues are orthogonal
3. The eigenvectors span the space