- #1
mmwave
- 647
- 2
I'm trying to show the relation between L^2 and Lz where L is total angular momentum and Lz is the z component.
Given f is an eigenfunction of both L^2 and Lz
L^2f = [lamb] f Lz f = [mu] f and L^2 = Lx^2 + Ly^2 + Lz^2 then
<L^2> = < Lx^2 + Ly^2 + Lz^2> = <Lx^2> + <Ly^2> + <Lz^2>
<L^2> = [inte]f* L^2 f dr = [inte]f* [lamb] f dr
= [lamb] [inte]f*fdr = [lamb] <f> = [lamb]
<Lz> = [inte]f* Lz^2 f dr = [inte]f* [mu]^2 f dr = [mu]^2 [inte]f*f dr
= [mu]^2
Now
<lz^2> = <L^2> - <Lx^2> - <Ly^2> substituting gives
[mu]^2 = [lamb] - <Lx^2> - <Ly^2>
or [mu] squared is less than or equal to [lamb] with equality only when <Lx^2> = <Ly^2> = zero
I would like to know if the above is correct and is this just another way of saying that Lz^2 <= to L^2 ?
Given f is an eigenfunction of both L^2 and Lz
L^2f = [lamb] f Lz f = [mu] f and L^2 = Lx^2 + Ly^2 + Lz^2 then
<L^2> = < Lx^2 + Ly^2 + Lz^2> = <Lx^2> + <Ly^2> + <Lz^2>
<L^2> = [inte]f* L^2 f dr = [inte]f* [lamb] f dr
= [lamb] [inte]f*fdr = [lamb] <f> = [lamb]
<Lz> = [inte]f* Lz^2 f dr = [inte]f* [mu]^2 f dr = [mu]^2 [inte]f*f dr
= [mu]^2
Now
<lz^2> = <L^2> - <Lx^2> - <Ly^2> substituting gives
[mu]^2 = [lamb] - <Lx^2> - <Ly^2>
or [mu] squared is less than or equal to [lamb] with equality only when <Lx^2> = <Ly^2> = zero
I would like to know if the above is correct and is this just another way of saying that Lz^2 <= to L^2 ?