Quantum Mechanics for Mathematicians

What is the initial wavefunctionψ(q,0)?
Show that at later times|ψ(q,t)|^2 is peaked about a point that moves with
velocity~mk.

Problem 3:
Show that the limit asT→0 of the propagator

U(T,qT−q 0 ) =

√

m

i 2 πT

e−

i 2 mT(qT−q^0 )^2

is aδ-function distribution.

Problem 4:
Use the Cauchy integral formula method of section 12.6 to derive equation
12.9 for the propagator from equation 12.13.

Problem 5:
In chapter 10 we described the quantum system of a free non-relativistic
particle of massminR^3. Using tensor products, how would you describe a
system of two identical such particles? Find the Hamiltonian and momentum
operators. Find a basis for the energy and momentum eigenstates for such a
system, first under the assumption that the particles are bosons, then under the
assumption that the particles are fermions.

B.7 Chapters 14 to 16

Problem 1:
Consider a particle moving in two dimensions, with the Hamiltonian function

h=

1

2 m

((p 1 −Bq 2 )^2 +p^22 )

• Find the vector fieldXhassociated to this function.


• Show that the quantities

p 1 and p 2 −Bq 1

are conserved.

• Write down Hamilton’s equations for this system and find the general
  solutions for the trajectories (q(t),p(t)).

This system describes a particle moving in a plane, experiencing a magnetic
field orthogonal to the plane. You should find that the trajectories are circles
in the plane, with a frequency called the Larmor frequency.

Problem 2:
Consider the action of the groupSO(3) on phase spaceR^6 by simultaneously
rotating position and momentum vectors.