G25.2666: Quantum Mechanics II

Problem set #3
Due: March 1, 2005

1.
Consider three spin-1/2 particles with spins , and . Define as the total spin. Find a set of eigenvectors for and in terms of the direct products of the spin eigenstates, , , of the individual particles.

2.
Consider a system composed of two spin-1/2 particles fixed in space described by a Hamiltonian of the form

where and are the -components of the spin operators and for the two particles, respectively.

a.
Suppose the initial state vector is a triplet state

At time, , is measured. What values can be found and with what probabilities?

b.
Suppose the initial state vector is the singlet state

At time, , is measured. What values can be found and with what probabilities?

c.
Can this Hamiltonian ever evolve a triplet state into a singlet state at any time ? Can you write down a Hamiltonian that can effect such an evolution?

3.
Consider a single quantum mechanical particle in one dimension with position and momentum operators and , respectively. The Hamiltonian of the system is

Recall that the time evolution of the system can be studied equally well using a stationary state picture (the Heisenberg picture) in which the operators and evolve in time accoring to Heisenberg's equations of motion:

The general solution to Heisenberg's equations will be

where and are the usual Schrödinger operators for position and momentum, respectively.

Consider a set of Hermitian operator-valued functions in terms of the Schrödinger operators, and that satisfy a Lie algebra:

where are the structure constants of the group and are, therefore, assumed known. Such a set of functions forms an operator basis in terms of which any Hermitian operator-valued function can be expanded according to:

where is a set of complex expansion coefficients.

a.
Express the Hamiltonian and solutions to Heisenberg's equations of motion as expansions with respect to the operator functions . Which sets of expansion coefficients will be functions of time?

b.
Starting from Heisenberg's equations, derive equations of motion for the time-dependent coefficients in terms of the structure constants of the Lie group.

c.
What are the initial conditions on your equations of motion?

d.
Of the conditions:

which of these impose additional conditions on the equations of motion and what, if any, are these conditions?