n= 0, 1 , 2 , 3 ...Theangular momentum must be quantized in units of ̄h.
This will prove to be true for 3 dimensions too, however, the 3 components of angular momentum
do not commute with each other, leading to all kinds of fun.
9.7 Derivations and Computations
9.7.1 Probability Flux for the Potential Step*
The probability flux is given by
j(x,t) =̄h
2 mi[
ψ∗∂ψ
∂x−
∂ψ∗
∂xψ]
We can save some effort by noticing that this contains an expressionminus its complex conjugate.
(This assures that term in brackets is imaginary and the flux is then real.)
j=̄h
2 im[
u∗du
dx−
du∗
dxu]
=
̄h
2 im[
u∗du
dx−CC
]
Forx < 0
j =̄h
2 im[(e−ikx+R∗eikx)(ikeikx−ikRe−ikx)−CC]j =
i ̄hk
2 im[1−Re−^2 ikx+R∗e^2 ikx−R∗R] +CCj = [1−|R|^2 ]̄hk
mThe probability to be reflected is the reflected flux divided by the incident flux. In this case its easy
to see that its|R|^2 as we said. Forx > 0
j=|T|^2̄hk′
mThe probability to be transmitted is the transmitted flux divided by the incident flux.
|T|^2
̄hk′
mm
̄hk=
4 k^2
(k+k′)^2k′
k=
4 kk′
(k+k′)^2again as we had calculated earlier.
9.7.2 Scattering from a 1D Potential Well*
V(x) =
0 x <−a
−V 0 −a < x < a
0 x > a