Introduction to Quantum Mechanics, Fall 2014
Problem Set 6
Due Tuesday, November 18
Problem 1: Consider a particle moving in two dimensions, with the Hamiltonian function
1
((p1 − Bq2 )2 + p22 )
2m
• Find the vector field Xh associated to this function.
h=
• Show that the quantities
p1 and p2 − Bq1
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 group SO(3) on phase space R6 by
simultaneously rotating position and momentum vectors.
• For the three basis elements lj of so(3), show that the momentum map
gives functions µlj that are just the components of the angular momentum.
• Show that the maps
lj ∈ so(3) → µlj
give a Lie algebra homomorphism from so(3) to the Lie algebra of functions
on phase space (with Lie bracket on such functions the Poisson bracket).
Problem 3: In the same context as problem 2, compute the Poisson brackets
{µlj , qk }
between the angular momentum functions µlj and the configuration space coordinates qj . Compare this calculation to the calculation of
π 0 (lj )ek
for π the spin-1 representation of SO(3) on R3 (the vector representation).
Problem 4: Consider the symplectic group Sp(2d, R) of linear transformations
of phase space R2d that preserve Ω.
1
• Consider the group of linear transformations of phase space R2d that act
in the same way on positions and momenta, preserving the standard inner
products on position and momentum space. Show that this group is a
subgroup of Sp(2d, R), isomorphic to O(d).
• Using the identification between Sp(2d, R) and matrices from the notes,
which matrices give the subgroup above?
• Again in terms of matrices, what is the Lie algebra of this subgroup?
• Identifying the Lie algebra of Sp(2d, R) with quadratic functions of the
coordinates and momenta, which such quadratic functions are in the Lie
algebra of the SO(d) subgroup?
• Consider the function
d
1X 2
(q + p2j )
2 j=1 j
What matrix does this correspond to as an element of the Lie algebra of
Sp(2d, R)? Show that one gets an SO(2) subgroup of Sp(2d, R) by taking
exponentials of this matrix. Is this SO(2) a subgroup of the SO(d) above?
2