Physics 401  Homework #7
1)
Orthogonality of the stationary states of the harmonic oscillator (three points).
Since energy is an observable quantity, it is represented by a Hermitian operator (
H
ˆ
),
and this guarantees that the energy eigenstates (or stationary states) are orthogonal.
We would like to demonstrate that the stationary states are, in fact, orthogonal, for a few
specific cases. To do this, show that the ground state, first excited state, and second
excited state of the harmonic oscillator have zero overlap, working in the position basis.
In other words, prove that
0
2
0
2
1
1
0
using the spatial wavefunctions of
these states:
2
/
4
/
1
0
2
!
2
1
)
(
e
H
n
m
x
n
n
n
,
x
m
0
2
4
2
1
2
2
1
0
H
H
H
Hint: By examining the evenodd properties of the integrands, you can easily show that
some of these integrals are zero without any real work.
2) (3 points) Suppose we prepare a simple harmonic oscillator with the following
wavefunction:
2
/
)
(
9
exp
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 Fall '11
 HALL
 Atom, Energy, Work, Quantum Physics

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