Physics 451
Quantum mechanics I
Fall 2012
Oct 4, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework
• HW # 10 extended until Friday Oct 5 by 7pm
Pb 2.33, 2.34, 2.35
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
Scattering state
E 0
Outside the well
For
d 2
2mE
2





k

2
2
dx
x  a
  Ae  Be
ikx
For
 ikx
xa
  Fe
ikx
Ch 2.6
Quantum mechanics
The finite square well
Scattering state
For
a  x  a
E 0
V ( x)  V0
d 2
2m
2


(
E

V
)



l

0
2
2
dx
General solution
  Ce  De
ilx
 ilx
l
2m( E  V0 )
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
Continuity at boundaries
V(x)
x
C,D
A
F
B
-V0
at x = +a
at x = -a
Continuity of
Continuity of

d
dx
Feika  Ceila  De  ila
Aeika  Beika  Ce ila  Deila



ik Ae ika  Beika  il Ce  ila  Deila


ikFeika  il Ceila  De  ila

Quantum mechanics
The finite square well
Continuity at boundaries
V(x)
x
A
C,D
F
B
-V0
Finally
i

 2ik
2
2
A   cos(2la ) 
k

l
sin(2
la
)


e F
2kl




sin(2la) 2
Bi
l  k2 F
2kl
Quantum mechanics
The finite square well
Scattering state
V(x)
x
A
F
B
-V0
Coefficient of transmission
F
T
A
2
1
(l 2  k 2 )2
2
 1
sin
 2la 
2 2
T
4k l
V0 2
1

2  2a
 1
sin 
2m  E  V0  
T
4 E  E  V0 


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
Quiz 14
We have seen that the coefficient of transmission oscillates with energy,
and that the well becomes ‘transparent” for a particle in a scattering state
when its energy equals specific values En.
Similarly, we can show that the coefficient of reflection oscillates and
the well becomes like a perfect wall, so the particle is totally reflected
for some other specific values of energy En’.
A. True
B. False
Ch 2.5
Quantum mechanics
Transmission versus energy
Delta function well
Transmission coefficient
T

1
1  m 2 / 2 2 E


V0 2
T  1 
sin 2  E  V0
 4E  E  V 
0






1
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
V0
Pb. 2.35
Pb. 2.34
With a different
definition for the
transmission coefficient
Analogy to
physical potentials