(10.21)
This term equals 1 when l =L, 0.5 when l =L/2, and 0.05 when l =0.2L.
Orthogonality
One of the special properties of eigenfunctions is that the solutions are
orthogonal. This means that although the solutions are related to each
other their overlap is zero; that is
(10.22)
Average or expectation value
In quantum mechanics, one can only determine the expectation, or aver-
age, value of a parameter. For example, for this problem let’s calculate
first the average momentum W:
(10.23)
(10.24)
(10.25)
The average momentum is zero because the particle will travel both to the
right and the left with equal probability, resulting in no net momentum.
The mean square of the momentum W^2 does not average to zero and is
calculated as:
(10.26)
(10.27)
(10.28)
W^2
22
22
== 4 2hn
Lh
λW
2 Z^2
02
2
= sin sin⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
∫ ⎟⎟
Lnx
Ln
Ln
L
ππ ππx
Lx⎛
⎝
⎜⎜
⎞
⎠
⎟⎟dW
2 Z
02
= () ()⎛
⎝
⎜⎜
⎞
⎠
∫ψ ⎟⎟
∂
∂
*dx ψ
ixxxLW
ZZ
==∫sin cos sin
2
02
iLn
Lnx
Lnx
Lx
i
Lnx
L
πππ π
d
LLL
⎛
⎝⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
00
W
Z
= sin sin⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
∫ ⎟⎟
2
L 0
nx
Lixnx
L
π∂
∂π
LLx⎛
⎝
⎜⎜
⎞
⎠
⎟⎟dW
Z
= () ()
⎛
⎝
⎜⎜
⎞
⎠
∫ψ ⎟⎟
∂
∂
*dx ψ
ixxxL0∫ ψψ τnn* ′d =
allspace
0
=−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
sin1
2
2
L
lL
nnl
π Lπ=−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢
⎢
⎤
sin21
2
2
L 2
xnx
LL
nπ
π ⎦⎦⎥
⎥
0l