4.2 One-Dimensional System
Let us apply the aforementioned general description to individual cases of Chaps. 1–
3.
Example 4.1: A Particle Confined in a Square-Well Potential This example was
treated in Chap. 1. As before, we assume that a particle (i.e., electron) is confined in a
one-dimensional system [ÀL x L (L > 0)].
We consider the optical transition from the ground state ϕ 1 (x) to the first excited
state ϕ 2 (x). Here, we put L ¼ π/2 for convenience. Then, the normalized coherent
state ψ(x) is described as
ψ x, t
ð Þ ¼
1
ffiffi ffi
2
p ϕ 1 x
ð Þ exp ÀiE 1 t=h
ð
Þþϕ 2 x
ð Þ exp ÀiE 2 t=h
ð
Þ
½
Š ,
ð4:8Þ
where we put c 1 ¼ c 2 ¼
1 ffiffi
2
p in (4.2). In (4.8), we have
ϕ 1 x
ð Þ ¼
ffiffiffi
2
π
r
cos x and ϕ 2 x
ð Þ ¼
ffiffiffi
2
π
r
sin 2x:
ð4:9Þ
Following (4.3), we have a following real function called a probability distribution
density:
ψ
à x, t
ð Þψ x, t
ð Þ ¼
1
π
cos
2 x þ sin
2 2x þ sin 3x þ sin x
ð
Þ cos ωt
Â
Ã
,
ð4:10Þ
where ω is given by (4.4) as
ω ¼ 3h=2m,
ð4:11Þ
where m is a mass of an electron. Rewriting (4.10), we have
ψ
à x, t
ð Þψ x, t
ð Þ ¼
1
π
1 þ
1
2
cos 2x À cos 4x
ð
Þ þ sin 3x þ sin x
ð
Þ cos ωt
h
i
: ð4:12Þ
Integrating (4.12) over À
π
2 ,
π
2
Â
Ã
, a contribution from only the first term is nonvanishing to give 1, as anticipated (because of the normalization).
Putting t ¼ 0 and integrating (4.12) over a positive domain 0,
π
2
Â
Ã
, we have
Z π=2
0
ψ
à x, 0
ð Þψ x, 0
ð Þdx ¼
1
2
þ
4
3π
% 0:924:
ð4:13Þ
Similarly, integrating (4.12) over a negative domain À
π
2 , 0
Â
Ã
, we have
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4 Optical Transition and Selection Rules