gjL z f
h
i¼
ħ
i
g 2π
ð Þ
à f 2π
ð Þ À g 0
ð Þ
à f 0
ð Þ
½
þ
Z 2π
0
ħ
i
∂
∂ϕ
g x
ð Þ
! Ã
f x
ð Þdϕ:
ð1:154Þ
Requiring an arbitrary function f to satisfy a BC f(2π) ¼ f(0), we reach
gjL z f
h
i¼ L z gj f
h
i:
ð1:155Þ
Note that we must have the above BC, because ϕ ¼ 0 and ϕ ¼ 2π are spatially the
same point. Thus, we find that L z is Hermitian as well on this condition.
On the basis of aforementioned argument, let us proceed to quantum-mechanical
studies of a harmonic oscillator. Regarding the angular momentum, we will study
their basic properties in Chap. 3.
Reference
1. Møller C (1952) The theory of relativity. Oxford University Press, London
30
1 Schrödinger Equation and Its Application
h
i¼
ħ
i
g 2π
ð Þ
à f 2π
ð Þ À g 0
ð Þ
à f 0
ð Þ
½
þ
Z 2π
0
ħ
i
∂
∂ϕ
g x
ð Þ
! Ã
f x
ð Þdϕ:
ð1:154Þ
Requiring an arbitrary function f to satisfy a BC f(2π) ¼ f(0), we reach
gjL z f
h
i¼ L z gj f
h
i:
ð1:155Þ
Note that we must have the above BC, because ϕ ¼ 0 and ϕ ¼ 2π are spatially the
same point. Thus, we find that L z is Hermitian as well on this condition.
On the basis of aforementioned argument, let us proceed to quantum-mechanical
studies of a harmonic oscillator. Regarding the angular momentum, we will study
their basic properties in Chap. 3.
Reference
1. Møller C (1952) The theory of relativity. Oxford University Press, London
30
1 Schrödinger Equation and Its Application