11.1. THEANGULARMOMENTUMCOMMUTATORS 173
where
[L ̃z,V(r)] = x[py,V(r)]−y[px,V(r)]= −i ̄h(
x∂V
∂y−y∂V
∂x)= −i ̄h(
x∂V
∂r∂r
∂y−y∂V
∂r∂r
∂x)= −i ̄h(
x∂V
∂ry
r−y∂V
∂rx
r)= 0 (11.9)
and
[L ̃z,p ̃^2 ] = [(xp ̃y−yp ̃x),p ̃^2 x+p ̃^2 y+p ̃^2 z]
= [xp ̃y,p ̃^2 x]−[yp ̃x,p ̃^2 y]
= p ̃y[x,p ̃^2 x]−p ̃x[y,p ̃^2 y]
= − 2 i ̄hp ̃yp ̃x+ 2 i ̄hp ̃xp ̃y
= 0 (11.10)Thisprovesthat
[L ̃z,H ̃]= 0 (11.11)
Similarstepsshowthat
[L ̃x,H ̃]=[L ̃y,H ̃]= 0 (11.12)
whichmeansthatinacentralpotential,angularmomentumisconserved.
However,sincetheorderofrotationsaroundthex,y,andz-axesareimportant,
theoperatorsL ̃x, L ̃y, L ̃zcannotcommuteamongthemselves. Forexample,
[L ̃x,L ̃y] = [(yp ̃z−zp ̃y),(zp ̃x−xp ̃z)]
= [yp ̃z,zp ̃x]+[zp ̃y,xp ̃z]
= yp ̃x[p ̃z,z]+xp ̃y[z,p ̃z]
= i ̄h(−yp ̃x+xp ̃y)
= i ̄hL ̃z (11.13)Altogether,
[L ̃x,L ̃y] = i ̄hL ̃z
[
L ̃y,L ̃z]
= i ̄hL ̃x
[
L ̃z,L ̃x]
= i ̄hL ̃y (11.14)Theseangularmomentumcommutatorsarethefundamentalequationsthatwillbe
appliedinthislecture.Theywillbeusedtodetermineboththeeigenvaluesandthe