Download Problem Set 12

Introduction to Quantum Mechanics, Spring 2015
Problem Set 12
Due Tuesday, March 24
Problem 1: Consider the Pauli-Schrödinger equation in three space dimensions
discussed in class.
• Find all the solutions to this equation, and show how the momentum
operator and space-translations act on solutions.
• Show explicitly how the double-cover of the Euclidean group acts on the
space of solutions.
• Find a solution of the equation that is a helicity eigenvector (eigenvector
of J · P) as well as a momentum eigenvector with momentum only in the z
direction. What happens to this solution when you act on it by the double
cover of the rotations about the x-axis? Find all the other solutions one
gets by acting on this by the double-cover of the Euclidean group.
Problem 2: Compute the propagator
G(x0 , t0 , x, t) = h0|ψ̂(x0 , t0 )ψ̂ † (x, t)|0i
for the free non-relativistic particle of mass m (for t0 > t). First do this in
momentum space, showing that
2
k
(t0 −t)
e 0 , t0 , k, t) = e−i 2m
G(k
δ(k 0 − k)
then Fourier transform to find the position space result
G(x0 , t0 , x, t) = (
m
1 −
m
(x0 −x)2
) 2 e i2(t0 −t)
0
i2π(t − t)
Show also that
lim G(x0 , t0 , x, t) = δ(x0 − x)
t→t0
Problem 3: Show that if one takes the quantum field theory Hamiltonian
operator to be
∞
1 d2
b
ψb† (x)(−
+ V (x))ψ(x)dx
2m dx2
−∞
Z
b =
H
the field operators will satisfy the conventional Schrödinger equation for the
case of a potential V (x).
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Problem 4: A quantum system corresponding to indistinguishable particles
interacting with each other with an interaction energy v(x − y) (where x, y are
the positions of the particles) is given by adding a term
Z
Z
1 ∞ ∞ b†
b ψ(x)dxdy
b
ψ (x)ψb† (y)v(y − x)ψ(y)
2 −∞ −∞
to the free particle Hamiltonian. Just as the free-particle Hamiltonian has an
expression as a momentum space integral involving products of annihilation and
creation operators, can you write this interaction term as a momentum space
integral involving products of annihilation and creation operators (in terms of
the Fourier transform of v(x − y)?
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