Chem 5314 homework #3 – due Oct. 13, 2011
Problem 1
– harmonic oscillator wavefunctions
In class, we found that the stationary states of the 1d harmonic oscillator have the form
ψ
n
=
A
n
(
x
n
+?
x
n

2
+?
x
n

4
+
· · ·
)
e

αx
2
(1)
where
A
n
is a normalization constant, the polynomial in brackets ends with
x
or a constant
if
n
is respectively odd or even, and where
α
=
mω
2
~
(2)
We did not derive a general formula for the coefficients (
i.e.
the ?’s) in the polynomial.
These
could
, although its not very practical, be determined by
orthogonality
.
For all of
question 1, express all your answers and do all your work in terms of the parameter
α
only.
a)
In particular, the second excited state
ψ
2
has the form
ψ
2
=
A
2
(
x
2
+
c
)
e

αx
2
(3)
Find the constant
c
by requiring that
ψ
2
be orthogonal to the ground state
ψ
0
.
b)
ψ
2
is also orthogonal to the first excited state
ψ
1
. Why? (hint: symmetry)
c)
Determine the normalization constant
A
0
for the ground state.
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 Fall '11
 Nielsen
 Physical chemistry, pH, Uncertainty Principle, ground state, ψn

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