Sample questions for PH320 test 3

7. Suppose we represent the kets |1〉 and |2〉 by
( )
1 1
√
2 −1
and
1
√
2
( 1
1
)
respectively. Then what is |1〉〈1| + |2〉〈2|
8. What approximation do we need to make to consider the ammonia molecule
NH 3 a two state system Would this be possible with the harmonic oscillator
9. Sketch the two lowest energy wavefunctions for NH 3 assuming we’ve approximated
the potential as a double-square-well.
10. What property of the wavefunctions tells you which has lowest energy
11. We know that any wavefunction ψ(x, 0) can be expanded in the normalized
eigenstates:
∞∑
ψ(x, 0) = B n ψ n (x).
Show that
ψ(x, t) =
n=0
∞∑
C n e −iEnt/¯h ψ n (x)
n=0
where C n and B n are some coefficients to be determined.
12. Consider a harmonic oscillator state:
|ψ〉 =
1 (
√ |1〉 + e iν |2〉 )
1 + λ 2
Find the values of λ and ν for which this state is normalized.
13. Prove that [Â, B]∗ = −[A, B] for self-adjoint operators.
14. Show that for any differentiable function f
[ ˆP , f(ˆx)] = −i¯hf ′ (ˆx)
15. Show that in classical mechanics the equations of a 1-D harmonic oscillator
can be written in the form:
Deduce from this that
d
(p + imωx) = iω(p + imωx).
dt
p(t) + imω(t) = e iωt [p(0) + imωx(0)]
2