ψ m
h
j ψ n i ¼ δ mn :
ð2:53Þ
At the same time, for c n of (2.38) we get
c n ¼
1
ffiffiffiffi
n!
p :
ð2:54Þ
Notice here that an undetermined phase factor e
iθ (θ : real) is intended such that
c n ¼
1
ffiffiffiffi
n!
p e
iθ
:
But, e
iθ is usually omitted for the sake of simplicity. Thus, we have constructed a
series of orthonormal eigenfunctions | ψ n i.
Furthermore, using (2.41) and (2.54) we get
a j ψ n i ¼
ffiffi ffi
n
p j ψ nÀ1 i:
ð2:55Þ
From (2.36), we get
H j ψ n i ¼ E 0 þ nħω
ð
Þjψ n i:
ð2:56Þ
Meanwhile, from (2.21) we have
H j ψ n i ¼ ħωa
{ a þ E 0
À
Á j ψ n i:
ð2:57Þ
Equating RHSs of (2.56) and (2.57), we get
a
{ a j ψ n i ¼ n j ψ n i:
ð2:58Þ
Thus, we find that an integer n is an eigenvalue of a
{ a when it is evaluated with
respect to jψ n i. For this reason a
{ a is called a number operator. Notice that a
{ a is
Hermitian because we have
a
{ a
À Á { ¼ a
{ a
{
À Á { ¼ a
{ a,
ð2:59Þ
where we used (1.115) and (1.117).
Moreover, from (2.55) and (2.58),
a
{ a ψ n i ¼
ffiffi ffi
n
p
a
{
ψ nÀ1 i ¼ n j ψ n i:
ð2:60Þ
Thus,
40
2 Quantum-Mechanical Harmonic Oscillator