À
ħ
2
2m
∇
2 ψ ¼
p
2
2m
ψ
ð1:36Þ
and the following correspondence
À
ħ
2
2m
∇
2 $
p
2
2m
,
ð1:37Þ
where m is the mass of a particle.
Meanwhile, taking partial differentiation of (1.27) with respect to t, we obtain
∂ψ
∂t
¼ À
i
ħ
Eψ 0 e
i
p
ħ ∙ xÀ
E
ħ t
ð
Þ ¼ À
i
ħ
Eψ:
ð1:38Þ
That is,
iħ
∂ψ
∂t
¼ Eψ:
ð1:39Þ
As the above, we get the following relationship:
iħ
∂
∂t
$ E:
ð1:40Þ
Thus, we have relationships between c-numbers (classical numbers) and q-numbers
(quantum numbers, namely, operators) in (1.35) and (1.40). Subtracting (1.36) from
(1.39), we get
iħ
∂ψ
∂t
þ
ħ
2
2m
∇
2 ψ ¼ E À
p
2
2m
ψ:
ð1:41Þ
Invoking the relationship on energy
Total energy
ð
Þ¼ Kinetic energy
ð
Þ þPotential energy
ð
Þ ,
ð1:42Þ
we have
E ¼
p
2
2m
þ V,
ð1:43Þ
where V is a potential energy. Thus, (1.41) reads as
10
1 Schrödinger Equation and Its Application