PHY6938 Proficiency Exam Spring 2002
April 5, 2002
Modern Physics and Quantum Mechanics
1. Consider the Schr¨
odinger equation for the linear harmonic oscillator,

¯
h
2
2
m
d
2
ψ
dx
2
+
1
2
mω
2
x
2
ψ
=
Eψ ,
where
m
is the mass of the particle and
ω
is the angular frequency. The
wavefunction and energy of the ground and first excited state are given
by
ψ
GS
(
x
) =
s
α
√
π
exp(

α
2
x
2
/
2)
, E
GS
=
1
2
¯
hω
ψ
E
1
(
x
) =
s
α
2
√
π
2
αx
exp(

α
2
x
2
/
2)
, E
E
1
=
3
2
¯
hω ,
respectively, where
α
=
q
mω/
¯
h
.
A perturbation term
H
0
=

mω
2
xx
0
is
added to the Hamiltonian of the harmonic oscillator.
The following integral may be useful:
Z
∞
∞
dx x
2
exp(

x
2
) =
√
π/
2
(a) Calculate the transition matrix element
h
ψ
GS

H
0

ψ
E
1
i
.
h
ψ
GS

H
0

ψ
E
1
i
=

mω
2
x
0
α
2
s
2
π
Z
∞
∞
dx x
2
exp
h

α
2
x
2
i
=
[change variable,
x
0
=
αx
]
=

mω
2
x
0
s
2
π
1
α
Z
∞
∞
dx
0
x
0
2
exp
h

x
0
2
i
=

mω
2
x
0
s
2
π
1
α
√
π
2
=

mω
2
x
0
1
α
√
2
=

mω
2
x
0
¯
h
2
mω
!
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 Fall '1
 staff
 Mass, 2m, 0 2m, dx x2 exp, Modern Physics and Quantum Mechanics

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