- Use the commutator relation betweenAandA†to derive [H,A]. Now show thatAis the
lowering operator for the harmonic oscillatorenergy.
- Att= 0, aone dimensional harmonic oscillatoris in the stateψ(t= 0) =
√
3
4 u^0 +i
√
1
4 u^1.
Calculate the expected value ofpas a function of time.
- Att= 0, a harmonic oscillator is in a linear combination of then= 1 andn= 2 states.
ψ=
√
2
3
u 1 −
√
1
3
u 2
Find〈x〉and〈x^2 〉as a function of time.
- A 1D harmonic oscillator is in a linear combination of the energy eigenstates
ψ=
√
2
3
u 0 +
√
1
3
u 2.
Find〈x^2 〉.
