where
i.e. K = mw02, m is the mass of the particle and K is the
spring constant. In terms of
= x and
Eq. (1) becomes
The
time-independent Schrodinger equation for the harmonic oscillator, in terms of the
creation and annihilation operators is given by
in
terms defined in Eq. (4), i.e.
the equation
, in terms of Eq. (4), becomes
to be
positive since this amounts to multiplication by an arbitrary phase factor. Therefore
A0 is chosen such that , i.e.
becomes
can also be expressed in terms of x
as follows.
when
is expressed in terms of the variable x.
Therefore
Solving for B0 gives
be used in what
follows.
is
proportional to
. I.e.
. Note
that
. Thus
The arbitrary
phase factor will always be chosen to have the value of unity. Similarly is proportional to
. I.e.
Therefore
We therefore have
Once again we
follow the same procedure above for Eq. (21) but now for n = 3 to get
we have
where
acts on exp(-x2/2)
results in an nth order polynomial multiplied by exp(-x2/2),
i.e.
to
give