Page 121 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

112 Harmonic oscillator
^ y y ^ y y
N^ a jëiiˆ ^ a (N ‡ 1)jëiiˆ ^ a (ë ‡ 1)jëiiˆ (ë ‡ 1)^ a jëii (4:28)
where equations (4.23) and (4.26b) have been used. In this case we see that
^
y
y
^ a jëii is an eigenfunction of N with eigenvalue ë ‡ 1. The operator ^ a changes
^
the eigenstate jëii to an eigenstate of N with a higher value, ë ‡ 1, of the
eigenvalue. The energy of the oscillator is increased by "ù. Thus, the operator
y
^ a is called a raising operator or a creation operator.

Quantization of the energy
In the determination of the energy eigenvalues, we ®rst show that the
^
eigenvalues ë of N are positive (ë > 0). Since the expectation value of the
^
operator N for an oscillator in state jëii is ë,wehave
^
hëijNjëiiˆ ëhëijëiiˆ ë
^
The integral hëijNjëii may also be transformed in the following manner

^
2

hëijNjëiiˆhëij^ a ^ ajëiiˆ (^ aö )(^ aö ëi )dô ˆ j^ aö ëi j dô
y
ëi
The integral on the right must be positive, so that ë is positive and the
^
^
eigenvalues of N and H cannot be negative.
For the condition ë ˆ 0, we have

2
j^ aö ëi j dô ˆ 0
which requires that
^ aj(ë ˆ 0)iiˆ 0 (4:29)
For eigenvalues ë greater than zero, the quantity ^ ajëii is non-vanishing.
To ®nd further restrictions on the values of ë, we select a suitably large, but
otherwise arbitrary value of ë,say ç, and continually apply the lowering
operator ^ a to the eigenstate jçii, thereby forming a succession of eigenvectors
3
2
^ ajçii, ^ a jçii, ^ a jçii, ...
with respective eigenvalues ç ÿ 1, ç ÿ 2, ç ÿ 3, ... We have already shown
^
that if jçii is an eigenfunction of N, then ^ ajçii is also an eigenfunction. By
^
2
iteration, if ^ ajçii is an eigenfunction of N, then ^ a jçii is an eigenfunction, and
so forth, so that the members of the sequence are all eigenfunctions. Eventually
k
this procedure gives an eigenfunction ^ a jçii with eigenvalue (ç ÿ k), k being a
positive integer, such that 0 < (ç ÿ k) , 1. The next step in the sequence
would yield the eigenfunction ^ a k‡1 jçii with eigenvalue ë ˆ (ç ÿ k ÿ 1) , 0,
which is not allowed. Thus, the sequence must terminate by the condition
k
^ a k‡1 jçiiˆ ^ a[^ a jçii] ˆ 0