Physics 451
Quantum mechanics I
Fall 2012
Oct 1, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework this week:
• HW # 9 due Tuesday Oct 4 by 7pm
Pb 2.27, 2.29, 2.30, 2.31
• HW # 10 due Thursday Oct 7 by 7pm
Ch 2.5
Quantum mechanics
The delta function potential
V  x     x 
For
x0
d 2
2mE
 2 
2
dx
Continuity at boundaries

d
dx
is continuous
is continuous except where V is infinite
 d

 dx
2m



 0

2


Ch 2.5
Quantum mechanics
The delta function well
Bound state
E0
m  m x /  2
 x  
e

m 2
E
2 2
Pb 2.27 double delta well
2 boundaries, 4 conditions
Ch 2.5
Quantum mechanics
The delta function well
Scattering state
E 0
Travelling waves
 left  x   Ae  Be
ikx
 right  x   Feikx  Geikx
ikx
A
2mE
F
G
B
0
Continuity at boundary
k
(A,B, F,G) ?
x
Ch 2.5
Quantum mechanics
The delta function well
Scattering state
E 0
Travelling waves
 left  x   Ae  Be
ikx
 right  x   Feikx
ikx
A
k
2mE
F
B
x
0
Reflected wave
Transmitted wave
i
B
A
1  i
F
1
A
1  i

m
m 

2
k
2E
Ch 2.5
Quantum mechanics
The delta function well
E 0
Scattering state
 right  x   Feikx
 left  x   Aeikx  Beikx
A
k
2mE
F
B
x
0
Reflection coefficient
Transmission coefficient
B
2
R

A
1  2
F
1
T

A
1  2
1
R
1  2 2 E / m 2
T
2

2


1
1  m 2 / 2 2 E


m
m 

2
k
2E
Ch 2.5
Quantum mechanics
The delta function potential
Scattering state
E 0
Ch 2.5
Quantum mechanics
The delta function barrier
V  x     x 
Scattering state only
E 0
 right  x   Feikx
 left  x   Aeikx  Beikx
A
F
B
x
0
Reflection coefficient
1
R
1  2 2 E / m 2


Transmission coefficient
T

1
1  m 2 / 2 2 E

“Tunneling”
Ch 2.5
Quantum mechanics
Quiz 13
A particle can tunnel trough an infinite barrier
with some relatively small thickness
A. Yes
B. No
Ch 2.6
Quantum mechanics
The finite square well
V(x)
Scattering
states
a
-a
x
Bound states
E
-V0
Ch 2.6
Quantum mechanics
The finite square well
Continuity at boundaries
V(x)

x
-V0
X=-a
X=+a
d
dx
is continuous
is continuous
Ch 2.6
Quantum mechanics
The finite square well
Bound state
E0
For
x  a
For
xa
d 2
2mE
2




k

2
2
dx
  Aekx
  Be
 kx
Ch 2.6
Quantum mechanics
The finite square well
Bound state
For
E0
a  x  a
V ( x)  V0
d 2
2m
2


(
E

V
)



l

0
2
2
dx
General solution
  C sin  lx   D cos  lx 
l
2m( E  V0 )
Quantum mechanics
The finite square well
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
Pb 2.30 normalization
Quantum mechanics
The finite square well
Continuity at boundaries
V(x)
x
-V0
Continuity of
Continuity of

d
dx
Ae ka  D cos(la)
kAe ka   Dl sin(la)
k  l tan(la )
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
Bound states
• Wide, deep well
large
z0
• Shallow, narrow well
(large a or V0)
small
z0
(small a, V0)
V(x)
V(x)
x
x
-V0
n2 2 2
En  V0 
2m(2a)2
-V0
One bound state
Pb 2.29 odd solution
Pb 2.31 extrapolation to infinite delta well