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4.2 Quantum treatment 113
The only circumstance in which ^ a operating on an eigenvector yields the
value zero is when the eigenvector corresponds to the eigenvalue ë 0, as
k
shown in equation (4.29). Since the eigenvalue of ^ a jçii is ç ÿ k and this
eigenvalue equals zero, we have ç ÿ k 0 and ç must be an integer. The
minimum value of ë ç ÿ k is, then, zero.
Beginning with ë 0, we can apply the operator ^ a successively to j0ii to
y
form a series of eigenvectors
y2
y3
y
^ a j0ii, ^ a j0ii, ^ a j0ii, ...
^
with respective eigenvalues 0, 1, 2, ... Thus, the eigenvalues of the operator N
are the set of positive integers, so that ë 0, 1, 2, ... Since the value ç was
chosen arbitrarily and was shown to be an integer, this sequence generates all
^
the eigenfunctions of N. There are no eigenfunctions corresponding to non-
integral values of ë. Since ë is now known to be an integer n, we replace ë by n
in the remainder of this discussion of the harmonic oscillator.
The energy eigenvalues as related to ë in equation (4.25) are now expressed
in terms of n by
1
E n (n )"ù, n 0, 1, 2, ... (4:30)
2
so that the energy is quantized in units of "ù. The lowest value of the energy or
zero-point energy is "ù=2. Classically, the lowest energy for an oscillator is
zero.
Non-degeneracy of the energy levels
To determine the degeneracy of the energy levels or, equivalently, of the
^
eigenvalues of the number operator N, we must ®rst obtain the eigenvectors
j0ii for the ground state. These eigenvectors are determined by equation (4.29).
When equation (4.18a) is substituted for ^ a, equation (4.29) takes the form
d d
î j0ii î ö 0i (î) 0
dî dî
or
dö 0i
ÿî dî
ö 0i
This differential equation may be integrated to give
2
iá ÿ1=4 ÿî =2
ö 0i (î) ce ÿî =2 e ð e 2
where the constant of integration c is determined by the requirement that the
functions ö ni (î) be normalized and e iá is a phase factor. We have used the
standard integral (A.5) to evaluate c. We observe that all the solutions for the
ground-state eigenfunction are proportional to one another. Thus, there exists
