Physics 3316, Spring 2010
Basics of Quantum Mechanics
Problem Set 8
(Due in lecture, March 31, 2010)
Required Readings :
•
Griffiths Ch. 3
Key concepts:
1D potentials, Operator algebra
1
Resonant transmission
U
0
x=0
x=a
E
U(x)
x
Fig. 1:
Potential barrier
1.1
Quantum Amplitudes
Compute
as a function of the incoming wave vector
k
, the quantum amplitudes (
r
(
k
),
t
(
k
), respectively) for
scattering of leftincident particles from the potential of Figure 1.
Hints:
1. Your algebra will be simpler if you choose the solutions of the form
φ
1
(
x
) =
A
1
cos(
kx
) +
B
1
sin(
kx
),
φ
2
(
x
) =
A
2
cos(
k
′
(
x
−
a
)) +
B
2
sin(
k
′
(
x
−
a
)),
φ
3
(
x
) =
A
3
e
ik
(
x
−
a
)
, and then match the boundary conditions
first at
x
=
a
and then at
x
= 0. Making similar guesses “centering” all parts of the wave function on the
rightmost point in their interval, except for the rightmost part of the wave function which should be centered
on the leftmost part of its interval, and then solving the matching conditions in right to left order will make
the algebra in scattering problems. This approach is known as the
transfer matrix approach
and is a general
method for rending all physical problems in onedimension trivial in many advanced fields of physics.
1
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2 Hermitian operators
2
2. You should find
t
(
k
) =
1
cos(
k
′
a
)
−
i
2
(
k
k
′
+
k
′
k
)
sin(
k
′
a
)
,
for the transmission amplitude.
1.2
Probabilities for Transmission and Reflection
From your result compute
P
r
and
P
t
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 Spring '08
 FLANAGAN
 mechanics, Hilbert space, hermitian inner product

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