PHY 6645 Fall 2002 – Homework 8
Due by 5 p.m. on Wednesday, November 13. Partial credit will be available
for solutions submitted by 5 p.m. on Thursday, November 14. (Note the early
deadline due to Homecoming on November 15.)
Answer all questions. To gain maximum credit you should explain your reasoning and show
all working. Please write neatly and include your name on the front page of your answers.
1. What are the physically acceptable eigenvalues and eigenkets of the spatial displacement operator S(a)?
2. Consider a one-dimensional harmonic oscillator of mass m and frequency ω. Define
the state |bi = S(b)|0i, where |0i is the ground state of this problem, and S(b) is the
operator representing spatial displacement through b.
(a) Find [S(b), an ], where a is the annihilation operator for the harmonic oscillator.
You may find it helpful to use the standard identity [e.g., Merzbacher Eq. (3.54),
Sakurai Eq. (2.3.47)]
eαΩ Λe−αΩ = Λ +
α
α2
α3
[Ω, Λ] + [Ω, [Ω, Λ]] + [Ω, [Ω, [Ω, Λ]]] + . . . ,
1!
2!
3!
where Ω and Λ are arbitrary linear operators, and α is a complex number.
(b) Use the result from part (a) to express |bi as a linear combination of eigenstates
of the number operator. Make sure that the state function is unit-normalized.
(c) Use your result from (b) to verify explicitly that hb|X|bi = b.
(d) Show that |bi can be expressed as a coherent state of the form
exp(za† − 12 |z|2 )|0i,
where z is a complex number. (Coherent states were introduced in Homework 7.)
3. The motion of a particle in one dimension is described by the Hamiltonian
H = P 2 /2m − A sin ωt,
where A and ω are positive, real constants.
(a) Solve this problem by integrating the Schrödinger equation to find the Schrödingerpicture wave function ψS (x, t).
Hint: Use the method of separation of variables (Shankar p. 255).
(0)
(b) Reformulate the problem in the interaction picture, choosing HS = P 2 /2m and
(I)
HS = −A sin ωt. Find the wave function ψI (x, t) and calculate the matrix elements hx|XI (t)|x0 i and hx|PI (t)|x0 i.
4. A quantum mechanical state |ψi is known to be a simultaneous eigenstate of two
Hermitian operators A and B which anticommute:
AB + BA = 0.
What can you say about the eigenvalues of A and B for state |ψi? Illustrate your
point using the parity operator (which can be chosen to satisfy Π = Π−1 = Π† ) and
the momentum operator.
5. Ballentine Problem 13.7: Suppose that the Hamiltonian is invariant under time reversal: [H, T ] = 0. Show that, nonetheless, an eigenvalue of T is not a conserved
quantity.