130_notes.dvi

6.7.5 Commutator ofLxandLy

Angular momentum is defined by
~L=~r×~p.

So the components of angular momentum are

Lz=xpy−ypx

Lx=ypz−zpy

Ly=zpx−xpz.

We wish to compute [Lx,Ly] which has all the coordinates and momenta in it.

The only operators that do not commute are the coordinates and their conjugate momenta.

[x,y] = 0

[px,py] = 0

[pi,rj] =

̄h

i

δij

So now we just need to compute.

[Lx,Ly] = [ypz−zpy,zpx−xpz]

= [ypz,zpx]−[ypz,xpz]−[zpy,zpx] + [zpy,xpz]

= y[pz,z]px− 0 −0 +x[z,pz]py

=

̄h

i

(ypx−xpy) =i ̄hLz

It is not necessary (or wise) to use the differential operators anda wave function crutch to compute
commutators like this one.Use the known basic commutators when you can.

6.8 Sample Test Problems

1. The absolute square of a wave function for a free particle is givenas:

|ψ(x,t)|^2 =

√

a

2 π(a^2 +b^2 t^2 )

e−a(x−vgt)

(^2) /2(a (^2) +b (^2) t (^2) )
Find the expected value ofxas a function of time. Find the expected value ofx^2 as a function
of time. Compute the RMS x-width of this wave packet as a function of time.

2. Find the commutator [p,eik^0 x] wherek 0 is a constant and the second operator can be expanded
   aseik^0 x=

∑∞

n=0

(ik 0 x)n

n!.

3. Which of the following are linear operators?