Problems

The Hamiltonian for the positronium atom in the 1S state in a magnetic field B is, to a good approximation,

 ,

where 1 labels the electron and 2 the positron.  Here S1 and S2 are the spin operators for the two particles and A is a constant.

(a) Choose the z-axis along B.  Which of the following are constants of motion?

1. S12 and S22
2. S1z and S2z
3. S2
4. S=S1+S2
5. Sz=S1z+S2z

(b) Find the energy eigenvalues of H in terms of the constants A and .


 

An electron (charge -e, and magnetic moment m) moves through a magnetic field.

(a) Write the Hamiltonian for this particle (assume non-relativistic kinematics).  Give m for the electron in Bohr magnetons.

(b) Now ignore the charge and spatial motion of the particle.  Assume a constant magnetic field B0 in the y-direction.  Write the Schroedinger equation for the spin state vector, explicitly giving the Hamiltonian.  At time t=0 the spin state is  . By solving the time-dependent Schroedinger equation in which one defines show that the state at time t1 such that  is .

(c) Suppose that a neutron in this spin state passes between the two poles of a magnet.  If B is inhomogeneous and in the x-direction, find the probability that the neutron will be found in the spin down state (in the x-direction) after passing the gap between the poles of the magnet.


 

An atom with L=0 and S= is initially in its ground state with Sz =- .  A field  is turned on at t=0 so that the effective Hamiltonian may be taken to be

,

where H0 is the atomic Hamiltonian with H1 = 0. Thus both Sz= and Sz=- states have the same eigenvalue of H0 in the ground state.

(a) What is the probability P, to lowest order in H1, that the atom remains in the state |S=,Sz=- > at time t?

Here  , e is the magnitude of the electron charge, me is the electron mass, and c is the velocity of light in the vacuum.

(b) Under what conditions on t will P=0?


 

The Hamiltonian of a system of two spin particles may be written as 

,

where S1 and S2 are the spin matrices for particle 1 and 2 respectively.

(a) Show that H commutes with the square of the total spin operator and the z-component of the total spin.

(b) Find the eigenspinors and eigenenergies of this Hamiltonian.