Introduction to Quantum and Statistical Mechanics
Homework 6 Solution
Prob. 6.1
For an infinitewall with a small step potential defined as:
(
29
≥
∞
≤
≤
≤
≤
≤
∞
=
L
x
L
x
d
V
d
x
x
x
V
0
0
0
0
We would like to look at the formalism to find the groundstate energy
E
.
Assume that
E > V
0
.
(a)
Set up the eigenfunctions
for 0 ≤ x ≤ d
and
d ≤ x ≤ L
, respectively.
(10 pts)
(
29
≥
≤
≤
≤
≤
≤
=
L
x
L
x
d
x
k
B
d
x
x
k
A
x
x
0
sin
0
sin
0
0
2
1
ψ
where
2
1
2
F
mE
k
=
and
(
29
2
0
2
2
F
V
E
m
k

=
(b)
Derive the transcendental equation setup which can give the solution for the quantized
ground energy
E
.
No analytical solution for the transcendental equation needed.
(10 pts)
Matching the boundary condition at
d
and
L
for both
ψ
and
d
ψ
/dx
and cancelling out
A
and
B
by division, we obtain:
(
29
(
29
(
29
d
k
V
E
L
d
k
E
1
0
2
tan
tan

=

Prob. 6.2
For the simple harmonic oscillator described by the Hamiltonian:
)
(
)
(
2
1
)
(
2
2
2
0
2
2
2
x
E
x
x
m
x
dx
d
m
φ
φ
ϖ
φ
=
+

(a)
If the particle is at the ground state, use the direct integration to find
∆
x
and
∆
p
.
Check
your answer with Eq. (6.12). The integration table for Gaussian functions at
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 '09
 KAN
 Trigraph, 2m, 10 pts, 2 2m, Mω

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