Question

given an electron has orbital angular momentum and spin angular momentum, l and s respectively.

given the following relations:

[lx,ly] = ih(bar)lz , [ly,lz] = ih(bar)lx , [lz,lx] = ih(bar)ly

[sx,sy]= ih(bar) sz , [sy,sz] = ih(bar)sx , [sz, sy\ = ih(bar)sy

(a) prove first that for an arbitary operator A, B that AB=BA+[A,B]

(b) show that [lx,l^2] = 0 where l^2 = l*l = lx^2 + ly^2 + lz^2

(c) show that similarly, [ly,l^2]=0 and [lz,l^2]=0

(d)given that the total angular momentum operator, j, is defined as j=l+s show that

j^2 = jx^2 +jy^2 +jz^2 = l^2 + s^2 + 2l*s

if l*s = lxsx + lysy + lzsz

(e) show that it is also possible to find the simultaneous eigenfunction of j^2 and jz (show as much work as possible)

(f) is it then possible to determine simultanesou eigenfunction of j^2, jz and l^2? (show as much work as possible)

***Please answer ALL parts, not just a few of them***

*** most important parts that I am confused on is part e and f***

Answer 1