Physics 451
Quantum mechanics I
Fall 2012
Review 2
Karine Chesnel
Quantum mechanics
EXAM II
When: Tu Oct 23 – Fri Oct 26
Where: testing center
• Time limited: 3 hours
• Closed book
• Closed notes
• Useful formulae provided
Quantum mechanics
EXAM II
1. The delta function potential
2. The finite square potential
(Transmission, Reflection)
3. Hermitian operator, bras and kets
4. Eigenvalues and eigenvectors
5. Uncertainty principle
Quantum mechanics
Square wells and delta potentials
V(x)
Physical considerations
 incident  x   Aeikx
 reflected  x   Beikx
Scattering
States E > 0
 transmitted  x   Feikx
x
Symmetry considerations
Bound states
E<0
 even   x    even  x 
 odd   x    odd  x 
Quantum mechanics
Square wells and delta potentials
Continuity at boundaries
Delta functions

d
dx
is continuous
is continuous except where V is infinite
 d
D
 dx

2 ma


 0 

2
h

Square well, steps, cliffs…

d
dx
is continuous
is continuous
Quantum mechanics
The delta function potential
V  x   a  x 
d 2

 a ( x)  E
2
2m dx
2
For
x0
d 2
2mE
 2 
2
dx
Quantum mechanics
The delta function well
Bound state
E0
ma  ma x / h 2
 x  
e
h
ma 2
E
2h 2
Ch 2.5
Quantum mechanics
The delta function well/ barrier
V  x   a  x 
E 0
Scattering state
 right  x   Feikx
 left  x   Aeikx  Beikx
A
F
B
x
0
Reflection coefficient
1
R
1  2 2 E / ma 2


Transmission coefficient
T

1
1  ma 2 / 2 2 E

“Tunneling”
Quantum mechanics
The finite square well
Bound state
E0
Symmetry considerations
V(x)
The potential is even function about x=0
The solutions are either even or odd!
x
Ae kx
 even 
-V0
D cos  lx 
Ae  kx
Quantum mechanics
The finite square well
Bound states
2
 z0 
tan z     1
 z 
where
z  la
z0 
a
2mV0
Quantum mechanics
The finite square well
Scattering state
E 0
(2)
V(x)
(1)
(3)
+a
-a
x
A
C,D
F
B
-V0

Aeikx  Beikx
(1)
Ceilx  De  ilx
(2)
Feikx
(3)
Quantum mechanics
The finite square well
V(x)
A
B
F
x
-V0
Coefficient of transmission

V0 2
 2a

T  1 
sin 2 
2m  E  V0   
 4E  E  V 

 
0

1
The well becomes transparent (T=1)
2
2
when En  V0  n 2m(2a)2
2
Quantum mechanics
Formalism
Wave function
Operators
 
Hˆ   H ij 
Observables are Hermitian operators
Vector
Linear transformation
(matrix)
Q Q
†
Quantum mechanics
Eigenvectors & eigenvalues
For a given transformation T, there are “special” vectors for which:
T a  a
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 
Find the eigenvectors
 e1, e2 ,...eN 
Spectrum
Quantum mechanics
Hilbert space
Infinite- dimensional space
N-dimensional space

e1 , e2 , e3 ,... eN


1
,  2 ,  3 ...  n ...
b

Hilbert space: functions f(x) such as
f ( x) dx  
2
a

Inner product
f g 

f ( x )* g ( 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
The uncertainty principle
Finding a relationship between standard deviations
for a pair of observables
 A2 B 2
  A, B 

 2i





2
Uncertainty applies only for incompatible observables
Position - momentum
Dx Dp 
2
Quantum mechanics
The uncertainty principle
Energy - time
DE Dt 
Derived from the
Heisenberg’s equation
of motion
d Q
dt

i
2
Q
 H , Q 
t
Special meaning of Dt
Dt 
Q
d Q
dt
Quantum mechanics
The Dirac notation
Different notations to express the wave function:
• Projection on the energy eigenstates
   cn n eiE t /
n
n
• Projection on the position eigenstates    ( y, t ) ( x  y )dy
• Projection on the momentum eigenstates
    ( p, t )
1
eipx / dp
2
Quantum mechanics
The Dirac notation
Bras, kets
= inner product
= matrix (operator)
Operators, projectors
en en  projector on vector en