Expectation Values of Angular Momentum Operators

  • Thread starter KiwiBlack
  • Start date
  • #1
2
0

Homework Statement


Show that

< l,m | Lx2 - Ly2 | l,m > = 0

Homework Equations



L2 = Lx2 + Ly2 + Lz2

[ Lx, Ly ] = i [STRIKE]h[/STRIKE] Lz

[ L, Lz ] = i [STRIKE]h[/STRIKE] Lx

[ Lz, Lx ] = i [STRIKE]h[/STRIKE] Ly



The Attempt at a Solution



I tried substituting different commutation values in place of Lx and Ly, but I'm not reducing it any further. I also tried ladder operations, but my professor said they're not needed to solve the problem.
 

Answers and Replies

  • #2
196
22
Well I think ladder operators are fine, but there are some "shortcut" ways too:

You could probably get away with just saying that it's "obvious" that
[tex]\langle l,m \lvert L_x^2 \lvert l,m \rangle = \langle l,m \lvert L_y^2 \lvert l,m \rangle[/tex]
because an Lz eigenstate shouldn't know the difference between the x and y directions. If you wanted to make that idea precise, you could find out what happens to the eigenstate and to the angular momentum operators when you do a 90-degree rotation about the z-axis.
 

Related Threads on Expectation Values of Angular Momentum Operators

Replies
4
Views
6K
Replies
2
Views
3K
Replies
11
Views
3K
Replies
4
Views
4K
  • Last Post
Replies
2
Views
374
Replies
1
Views
2K
Replies
16
Views
12K
Replies
1
Views
3K
Replies
10
Views
5K
Replies
5
Views
2K
Top