(Angular momentum) Define the angular momentum operator by
L = r x p.
where r = (r1, r2, r3) and p = (p1, p2, p3).
(a) Show that
L = 2p3 - p23
L = 3p - 31
L3 = p - p2.
(1)
b) Using the commutation relation
[x, p] = h
(2)
show that
[Li, L] = hL3
[L2, L] = zhL1
[L3, L] = ihL2.
(3)
(c) Deduce further that
[Li, L2 + L3 + L3] = 0
(4)
for each i = 1, 2, 3.
(d) Show that if, in three dimensions,
H = p2 + Vr / 2m.
where r = x1^2 + x2^2 + x3^2 and p = p1^2 + p2^2 + p3^2, then L generates a symmetry of H for all i = 1, 2, 3, that is [L, H] = 0.