Physics 212 – Problem Set 11 – Spring 2010
1. Scattering in 1D.
(a) Consider a single potential step. The potential is zero from
x <
0 and a constant
V
0
for
x >
0. A particle
of mass
m
is incoming from
x <
0 with energy
E
= ¯
hω
.
i. Write the Schrodinger equation for
x <
0. Write the Schrodinger equation for
x >
0. Write expressions
for the general solution in the region where
x <
0 and in the region where
x >
0.
ii. Remembering that the particle is incoming from
x <
0, find the probability current for the
x <
0
region. Find the probability current for the
x >
0 region. From this find the probability of transmission
(=
j
trans
j
inc
) and the probability of reflection (=
j
ref
j
inc
) in terms of the relevant wave numbers and unknown
coefficients.
(b) The general solution for an arbitrary
V
(
x
) may be written Ψ(
x, t
) =

Ψ(
x, t
)

e
iS
(
x,t
)
/
¯
h
. Find the probability
current associated with this solution. Notice how the current depends upon the
phase
S
(
x, t
)
(c) Suppose we could write the general solution for a potential
V
(
x
) as Ψ(
x, t
) =
Ae
iS
(
x,t
)
/
¯
h
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 Spring '09
 mechanics, Electron, Energy, Mass, Schrodinger Equation, 2m, form factor

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