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4.2 Quantum treatment 111
^
y
N ^ a ^ a (4:21)
^
The adjoint of N may be obtained as follows
^
^
N (^ a ^ a) ^ a (^ a ) ^ a ^ a N
y
y
y y
y
y
y
^
where the relations (3.40) and (3.37) have been used. We note that N is self-
adjoint, making it hermitian and therefore having real eigenvalues. Equation
(4.19b) may now be written in the form
^
^
1
H "ù(N ) (4:22)
2
^
^
Since H and N differ only by the factor "ù and an additive constant, they
commute and, therefore, have the same eigenfunctions.
^
If the eigenvalues of N are represented by the parameter ë and the
corresponding orthonormal eigenfunctions by ö ëi (î) or, using Dirac notation,
by jëii, then we have
^
Njëii ëjëii (4:23)
and
^ ^ 1 1
Hjëii "ù(N )jëii "ù(ë )jëii E ë jëii (4:24)
2 2
^
Thus, the energy eigenvalues E ë are related to the eigenvalues of N by
E ë (ë )"ù (4:25)
1
2
The index i in jëii takes on integer values from 1 to g ë , where g ë is the
^
degeneracy of the eigenvalue ë. We shall ®nd shortly that each eigenvalue of N
is non-degenerate, but in arriving at a general solution of the eigenvalue
equation, we must initially allow for degeneracy.
^
From equations (4.20) and (4.21), we note that the product of N and either ^ a
or ^ a may be expressed as follows
y
^
^
y
y
y
N^ a ^ a ^ a^ a (^ a^ a ÿ 1)^ a ^ a(^ a ^ a ÿ 1) ^ a(N ÿ 1) (4:26a)
^
^
y
N^ a ^ a ^ a^ a ^ a (^ a ^ a 1) ^ a (N 1) (4:26b)
y
y
y
y
y
These identities are useful in the following discussion.
^
If we let the operator N act on the function ^ ajëii, we obtain
^ ^
N^ ajëii ^ a(N ÿ 1)jëii ^ a(ë ÿ 1)jëii (ë ÿ 1)^ ajëii (4:27)
where equations (4.23) and (4.26a) have been introduced. Thus, we see that
^
^ ajëii is an eigenfunction of N with eigenvalue ë ÿ 1. The operator ^ a alters the
^
eigenstate jëii to an eigenstate of N corresponding to a lower value for the
eigenvalue, namely ë ÿ 1. The energy of the oscillator is thereby reduced,
according to (4.25), by "ù. As a consequence, the operator ^ a is called a
lowering operator or a destruction operator.
^
Letting N operate on the function ^ a jëii gives
y
