We have already found the ground state wave function,
.
If we require the wave function to be properly normalized, i.e.
,
then
.
Therefore
.
We have previously shown that
,
with
.
We therefore have
,
or,
,
,
.
is the product of 
and a polynomial of degree n and
parity (-1)n called a Hermite polynomial.
.
Specifically,
.
.
with 
.
Link:
 | Harmonic oscillator wavefunctions |
Let us look at the Hermite polynomials in somewhat more detail. Consider a Gaussian
function, 
.
.
We write
.
This defines the Hermite polynomials.
.
The parity of the Hermite polynomials is (-1)n.
Hn(z) is a nth degree polynomial in z.
Proof:
This statement holds for n=1 and n=2.
Let 
. Then
.
If Hn-1(z) is a polynomial of degree n-1, the Hn(z) is a polynomial of degree n. The statement therefore holds for all n.
Consider F(z+l)=exp(-(z+l)2).
A Taylor series expansion, 
, yields
.
Let l’=-l. Then
.
. (Taylor series expansion)
is called the generating
function for the Hermite polynomials Hn(z).
We have already found
.
Other recurrence relations exist.
and combining the first two relations
.
[
is obtained by differentiating 
with respect to z,
, and equating terms of equal
power of l.
is obtained by
differentiating 
with respect to l.]
The Hermite polynomials Hn(z) are defined by
or 
.
The generating function of the Hermite polynomials is
.
The recurrence relations are
, 
,
and
.
The differential equation satisfied by the Hermite polynomials is
.
What do the Hermite polynomials have to do with the harmonic oscillator?
The energy eigenfunction of the harmonic oscillator are
,
or
.
The fn(x) are solutions to the time independent Schroedinger equation
.
.
.
Assume that fn(h)=hn(h)exp(-h2/2) where hn(h) is a polynomial of degree n. Then hn(h) satisfies the differential equation
.
hn(h) satisfies the
differential equation satisfied by the Hermite polynomials. Therefore 
.
The energy eigenfunction of the harmonic
oscillator therefore are
.
oscillator with 
is
.
The Hermite polynomials 
satisfy
the recurrence relations
.
The normalization constants Nn are given by
.
(a) Show that the matrix elements of x can be expressed as
.
(b) Derive a similar expression for the matrix elements of x2.
(c) The ladder operators for the harmonic oscillator have the following properties:
.
Derive the expressions for aT and a in terms of 
.
Hint: Start by investigating the action of the
operator 
.
| Solution:
(a)
  
We have
therefore
We have
therefore
|
, 
.
.
.
.
.
.
.
, 
.
.
, 
.