Sample questions for PH320 test 3

Sample questions for PH320 test 3
I will give you the normalized harmonic oscillator wavefunctions are:
ψ n (x) =
π−1/4
√
2n n!a e−x2 /(2a 2) H n (x/a)
where a = √¯h/(mω). H 0 = 1. H 1 = 2x. H 2 = −2 + 4x 2 . H 3 = −12x + 8x 3 .
The energy levels are:
E n = (n + 1/2)¯hω
1. Show that the Hamiltonian operator for the harmonic oscillator is invariant
under “reflection”: Ĥ(x) = Ĥ(−x).
2. Define an operator ˆR the action of which on a wavefunction is to replace
x by −x.
(a) Show that ˆR is a linear Hermitian operator.
(b) Show that the eigenvalues of ˆR are ±1. Hint: apply ˆR twice.
(c) Show that the corresponding eigenfunctions are either even or odd.
(d) Show that if Ĥ(x) = Ĥ(−x), then ˆR commutes with Ĥ
3. Write down the time-dependent Schrödinger equation for the 1-D harmonic
oscillator. Solve this equation in the limit that x → ±∞.
4. Concerning he classical and quantum 1-D harmonic oscillators:
(a) What is the ground state wavefunction for the 1-D quantum harmonic
oscillator
(b) What is the general solution to the classical harmonic oscillator of
mass m and spring constant k
(c) Use your solution to construct a probability density function for the
classical oscillator. Sketch this classical probability and |ψ 0 (x)| 2 for
the ground state in the QM case. Explain the differences.
(d) What would you expect to see if you plotted |ψ n (x)| 2 versus the
classical case for a very large value of n.
5. Consider the operator: e ilp/¯h where l is a small (strictly, infinitesimal)
distance. Show that:
e ilp/¯h xe −ilp/¯h = x + l
6. Show that the ground state matrix element of this operator is:
〈0|e −ilp/¯h |0〉 = e −mωl2 /4¯h .
1

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

16. Show that the eigenvalues of a Hermitian operator are real.
17. Show that the Hermitian conjugate (adjoint) of λ|φ〉〈ψ|† ˆB ∗
is λ ˆB †Â|ψ〉〈φ|.
18. If the Hamiltonian does not depend on time, show that the state of the
system at any time t, denoted by |ψ(t)〉 can be deduced from the state
vector |ψ(t 0 )〉 at some initial time using
where Û(τ) = e−iĤτ/¯h .
|ψ(t)〉 = Û(t − t 0)|ψ(t 0 )〉
19. Show that Û † = Û −1 . This property is called unitarity.
20. Suppose we represent the two-state NH 3 kets |ψ S 〉 and |ψ A 〉 by
( ) 1
0
and (
0
1
)
.
What is the matrix transformation that trasforms these to the semiclassical
left and right states: |ψ L 〉 and |ψ R 〉 by
3