This preview shows pages 1–2. Sign up to view the full content.
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.
Problem 2
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '11
 Nielsen
 Physical chemistry, pH

Click to edit the document details