Physics 341
Midterm Solutions
November 10
Do not use calculators, cell phones, or cheat sheets. All you need is a pen or pencil.
Please show your work on these sheets.
1
. Consider an electron of mass
m
in one dimension subject to a potential
V
(
x
) =
gδ
(
x
)
where
g
is a constant and
δ
(
x
) is the Dirac deltafunction.
This system approximates a
highly localized potential around the point
x
= 0.
(i) For what sign(s) of
g
is there a quantum mechanical bound state (not a scattering state)?
(ii) Please find a relation between the bound state energy
E
and
g
.
Solution:
(i) Think of the potential as a limit of Gaussians. The attractive potential is the upside
down Gaussian so you want
g <
0 for a bound state.
(ii) Away from
x
= 0, the potential is vanishing so the wavefunction is free particle. For
x <
0, we can take
ψ
=
A
1
e
k
1
x
which is normalizable for
k
1
>
0. For
x >
0, similarly
ψ
=
A
2
e

k
2
x
.
This should be an energy eigenstate and the energy is
E
=

~
2
2
m
(
k
1
)
2
=

~
2
2
m
(
k
2
)
2
so
k
1
=
k
2
=
k
with
k >
0.
Lastly, we need to patch the wavefunctions at
x
= 0. By definition of the deltafunction
if we integrate around
x
= 0, we find:
Z


~
2
2
m
ψ
00
+
gψ
(0) =
Z

Eψ
(
x
)
.
(1)
The left hand side integral becomes

~
2
2
m
(
ψ
0
( )

ψ
0
(

)) =
~
2
k
2
m
(
A
1
+
A
2
)
.
So
ψ
0
is a step
function just like your homework problem. That means
ψ
is continuous at
x
= 0 so
A
1
=
A
2
(integrate again around the junction to check this). The right hand side of (1) therefore
vanishes. This gives the relation:
~
2
k
m
+
g
= 0
.
This makes sense since
k >
0 for normalizability and
g <
0. If
g
had the opposite sign,
there would be no normalizable solution. The energy is therefore
E
=

g
2
m
2
~
2
.
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 Spring '10
 Sghy
 mechanics, Energy, Mass, Work, Heat, ground state, ground state energy

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