Physics 451
Quantum mechanics I
Fall 2012
Oct 5, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework tonight!
• HW # 10 due Friday Oct 5 by 7pm
2.33, 2.34, 2.35
Homework next week:
• HW # 11 due Tuesday Oct 9 by 7pm
2.38, 2.39, 2.41, A1, A2, A5, A7
• HW # 12 due Thursday Oct 11 by 7pm
Quantum mechanics
Square wells and delta potentials
V(x)
Scattering
States E > 0
V(x)
 ( x)
V(x)
x
+a
-a
x
x
 ( x)
Bound states
E<0
-V0
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 
Ch 2.6
Quantum mechanics
Square wells and delta potentials
Continuity at boundaries
Delta functions

is continuous
d
dx
is continuous except where V is infinite
 d
D
 dx

2 m


 0 

2
h

Square well, steps, cliffs…

d
dx
is continuous
is continuous
Ch 2.6
Quantum mechanics
Square wells and delta potentials
Finding a solution
Scattering states:
Find the relationship between transmitted wave
and incident wave
Transmission coefficient
Tunneling effect
Bound states
Find the specific values of the energy
Ch 2.6
Quantum mechanics
Square barrier
V(x)
E  V0
V0
E  V0
E  V0
-a
x
+a
Pb. 2.33
Phys 451
The finite square barrier
Scattering states
V(x)
A
Pb. 2.33
F
B
x
-V0
Coefficient of transmission
F
T
A
1
(l 2  k 2 )2
2
 1
sin
 2la 
2 2
T
4k l
for
E  V0
2
1
2
 1   a 
T
1
(k 2   2 ) 2
 1
sinh 2  2 a 
2 2
T
4k 
for
for
E  V0
E  V0
Ch 2.6
Quantum mechanics
“Step” potential and “cliff”
V(x)
V(x)
V0
x
x
Pb. 2.34
• Reflection coefficient R
• Different definition for the
transmission coefficient T
(use the probability current J)
V0
Pb. 2.35
Analogy to
physical potentials
Quantum mechanics
Need for
a formalism
Wave function
Operator
 
Hˆ   H ij 
Vector
Matrix
Quantum mechanics
Vectors
Physical space
Generalization (N-space)
k
• Addition
- commutative
- associative
• Scalar multiplication
• zero vector
j
• linear combination
i
• basis of vectors
Quantum mechanics
Inner Product
Physical space
Generalization (N-space)
k
“Inner product”
B
• Norm
q
• Orthogonality
A
• Orthonormal basis
j
• Schwarz inequality
“Dot product”
i
ab
2
 aa bb
Quantum mechanics
Matrices
Physical space
Generalization (N-space)
• Linear transformation
k
Matrix
A’
 
T  Tij
• Sum
A’’
• Product
q
• Transpose
A
i
j
Transformations:
- Multiplication
- rotation
- symmetry…
• Conjugate
 
~
T  T ji
T *  Tij * 
• Hermitian conjugate T †  Tij * 
• Unit matrix
• Inverse matrix
• Unitary matrix
Quantum mechanics
Quiz 15
A matrix is “Hermitian” if:
~
A. T  T
T* T
C. T *  T
D. T~ *  T
E. T~ *  T
B.
Quantum mechanics
Changing bases
Physical space
k
Generalization (N-space)
k’
Old basis
j’
j
Expressing same transformation T
in different bases
Same determinant
Same trace
i
i’
New basis
Quantum mechanics
Formalism
N-dimensional space: basis
Norm:
a 
e
1
, e2 , e3 ,... eN

a a
Operator acting on a wave vector:
Expectation value/ Inner product
T a  b
T  aT a
T  a T a  T †a a
For Hermitian operators:
a T b  a T b  Ta b