From a point of view of a mechanical system, mathematical formulation of the
classical harmonic oscillator resembles that of electromagnetic fields confined within
a cavity. We return this point later in Sect. 9.6.
2.2 Formulation Based on an Operator Method
Now let us return to our task to find quantum-mechanical solutions of a harmonic
oscillator. Potential V is given by
V q
ð Þ ¼
1
2
sq
2
¼
1
2
mω
2 q
2 ,
ð2:9Þ
where q is used for a one-dimensional position coordinate. Then, we have a classical
Hamiltonian H expressed as
H ¼
p
2
2m
þ V q
ð Þ ¼
p
2
2m
þ
1
2
mω
2 q
2
:
ð2:10Þ
Following the formulation of Sect. 1.2, the Schrödinger equation as an eigenvalue
equation related to energy E is described as
Hψ q
ð Þ ¼ Eψ q
ð Þ
or
À
ħ
2
2m
∇
2
ψ q
ð Þ þ
1
2
mω
2 q
2
ψ q
ð Þ ¼ Eψ q
ð Þ:
ð2:11Þ
This is a SOLDE and it is well known that the SOLDE can be solved by a power
series expansion method.
In the present studies, however, let us first use an operator method to solve the
eigenvalue equation (2.11) of a one-dimensional oscillator. To this end, we use a
quantum-mechanical Hamiltonian where a momentum operator p is explicitly
represented. Thus, the Hamiltonian reads as
H ¼
p
2
2m
þ
1
2
mω
2 q
2
:
ð2:12Þ
Equation (2.12) is formally the same as (2.10). Note, however, that in (2.12) p and
q are expressed as quantum-mechanical operators.
As in (1.126), we first examine an expectation value hHi of H. It is given by
2.2 Formulation Based on an Operator Method
33