234
Appendix F: The Harmonic Oscillator
Here the H n are the Hermite orthogonal polynomials that can be derived from Rodriguez’ formula
H n (q) = (−1)
n e
q
2
d
n
dq n e
−q
2
(F.6)
resulting in
H 0 (q) = 1
H 1 (q) = 2q
H 2 (q) = 4q
2
− 2
H 3 (q) = 8q
3
− 12q
H 4 (q) = 16q
4
− 48q
2
+ 12
...
(F.7)
The normalization factor is
N n =
1
√
2 n n!
mω
π
1/4 .
(F.8)
The eigenfunction χ n has n nodes and is an even or odd function of q or x − x e , the
parity being that of the quantum number n. The χ n obey a simple recursion formula
that can be expressed as a property of the Hermite polynomials:
H n+1 (q) = 2q H n (q) − 2n H n−1 (q) .
(F.9)
Alternatively, one can define the ladder operators
ˆ
a =
q + i ˆ
p
√
2
and ˆ
a
†
=
q − i ˆ
p
√
2
(F.10)
with the following properties:
ˆ
a χ n (q) =
√
n χ n−1 (q)
(F.11)
ˆ
a
†
χ n (q) =
√
n + 1 χ n+1 (q) .
(F.12)
From Eq. (F.10) one gets
q =
ˆ
a + ˆ
a
†
√
2
and ˆ
p =
ˆ
a − ˆ
a
†
√
2i
.
(F.13)
These relationships are quite useful to compute matrix elements of q, ˆ
p, x, ˆ
p x , as
well as their powers and products. For instance
χ n |x| χ n = x e δ n,n + χ n |x − x e | χ n = x e δ n,n +
mω
χ n |q| χ n (F.14)