Physics 741 – Graduate Quantum Mechanics 1
Solution Set M
1.
[20] A particle of mass
m
and energy
E
in two dimensions is incident on a plane
step function given by
()
0
0
if
0,
,
if
0.
x
xy
x
Q
VQQ
VQ
<
⎧
=
⎨
>
⎩
The incoming wave has wave function
(
)
in
,
ikx ky
xy e
ψ
+
=
for
x
< 0.
(a) [7] Write the Hamiltonian.
Determine the energy
E
for the incident wave.
Convince yourself that the Hamiltonian has a translation symmetry, and
therefore that the transmitted and reflected wave will share something in
common with the incident wave (they are all eigenstates of what operator?).
The Hamiltonian, of course, is just
22
2
,,
pp
HV
x
y
V
x
y
mm
x
y
+
⎛⎞
∂∂
=+
=
−
+
+
⎜⎟
⎝⎠
=
The incident wave will therefore have an energy
()()
( )
( )
22 2
2
2
2
,
kk
EH
i
k
i
k
V
x
y
ψψ
+
==
−
+
+
=
=
=
,
so that
2
Ek
k
m
=
.
The potential depends on
x
, but is independent of
y
.
Therefore, they are
eigenstates of the translation operator in the
y
direction,
( )
ˆ
Ta
y
, for any
a
.
Since this is a
continuous symmetry, we can consider small translations, and we define the momentum
operator in the
y
direction
P
y
in the usual way, and our state must be an eigenstate of this
operator as well.
Our wave must therefore take the form
( ) ( )
,.
y
ik y
x
ye
X
x
=
It remains only to find the function
X
(
x
).
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View Full Document(b) [7] Write the general form of the reflected and transmitted wave.
Use
Schrödinger's equation to solve for the values of the unknown parts of the
momentum for each of these waves (assume
22
0
2
x
km
V
>
=
).
If we plug our general solution into Schrödinger’s equation, we have
()
(
)
() ( ) ()
2
22 2
2
,,
2
,.
2
yy
y
y
ik y
ik y
ik y
xy
ik y
ik y
ik y
y
Ee
X x
e
X x
V x y e
X x
mx
y
kk
k
e Xx
e Xx Vxye Xx
mm
m
x
⎛⎞
∂∂
=−
+
+
⎜⎟
⎝⎠
+
∂
+
∂
=
=
=
=
The exponentials cancel everywhere, and some factors can be cancelled.
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 Spring '04
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 Physics, Energy, Mass, Fundamental physics concepts, Ik, ik y ik

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