11.7 Examples
11.7.1 Time Development Example
Start off in the state.
ψ(t= 0) =1
√
2
(u 1 +u 2 )In the Schr ̈odinger picture,
ψ(t) =1
√
2
(u 1 e−i(^32) ωt
+u 2 e−i
(^52) ωt
) =
1
√
2
e−i(^32) ωt
(u 1 +e−iωtu 2 )
We can compute the expectation value ofx.
〈ψ|x|ψ〉 =
1
2
√
̄h
2 mω
〈u 1 +eiωtu 2 |A+A†|u 1 +eiωtu 2 〉=
1
2
√
̄h
2 mω(
〈u 1 |A|u 2 〉e−iωt+〈u 2 |A†|u 1 〉eiωt)
=
1
2
√
̄h
2 mω(√
2 e−iωt+√
2 eiωt)
=
√
̄h
mωcos(ωt)In the Heisenberg picture
〈ψ|x(t)|ψ〉=1
2
√
̄h
2 mω〈ψ|e−iωtA+eiωtA†|ψ〉This gives the same answer with about the same amount of work.
11.8 Sample Test Problems
- Calculate the commutator [Lx,Lz] whereLx=ypz−zpyandLz=xpy−ypx. State the
uncertainty principle forLxandLz.
Answer
[Lx,Lz] = [ypz−zpy,xpy−ypx] =x[y,py]pz+z[py,y]px=̄h
i(−xpz+zpx) =i ̄h(xpz−zpx) =−i ̄hLy∆Lx∆Lz ≥
i
2〈[Lx,Lz]〉=
i
2(−i ̄h)〈Ly〉=
̄h
2〈Ly〉2.*A particle of massmis in a1 dimensional potentialV(x). Calculate the rate of change
of the expected values ofxandp, (d〈dtx〉andddt〈p〉). Your answer will obviously depend on the