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 .

• Solution:

(a) The positronium atom consists of two spin ½ particles.  We neglect the spatial motion of the particles and consider only spin interactions.  The first term in the Hamiltonian is the spin-spin interaction, and the second term is the interaction of the spins with an external magnetic field.

.

In the { |S,Sz> } basis the matrix of S1×S2 is diagonal, while the matrix of is diagonal in the {|++>,|+->,|-+>,|-->} basis.  Let us work in the {|++>,|+->,|-+>,|-->} basis.

Then the matrix of S1×S2 is

,

and the matrix of S1z - S2z is

.

The matrix of H therefore is

,

where .

To find out if an observable is a constant of motion, we have to find out if it commutes with the Hamiltonian H.  If it commutes, it is a constant of motion.

1. .

2. The matrix of S1z is  .

The matrix of S2z is  .

S1z, S2z, and H do not have a set of common eigenvectors.  Therefore .

3. The matrix of S2 is . .

4. The matrix of S1x +S2x is  .

The matrix of S1y +S2y is  .

The matrix of S1z +S2z is  .

.

S1+S2 is not a constant of motion.

5. Sz=S1z+S2z is a constant of motion.

(b) The eigenvalue of H are found from E=C or from
(-C+D-E)(-C-D-E)-4C2 =0

The value E=C is twofold degenerate.

E2+2CE+C2-D2-4C2=0.

.

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.

• Solution:

(a)  The intrinsic magnetic moment of a particle is  .

For an electron  , where  is the Bohr magneton  in Gaussian units.

In SI units the Hamiltonian of the particle is

,

where

, , .

(b)  .

The matrix of H is   .

The energy eigenvalues are found from  .

Let  .

Let .

.

.

.

(c) A neutron is in the state

.

It is passed through the analyzer, which measures the Sx component.  The probability that the neutron will be found in the spin down state is  .

An atom with L=0 and S=½ is initially in its ground state with Sz =-½ .  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=-½ 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=½,Sz=-½ > 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?

• Solution:

(a)  Assume that the energy of the twofold degenerate ground state is much lower than the energy of any excited state, and approximate the system by a two-level system.
Denote the two levels by |+> and |->. H0|+>=E0|+>, H0|->=E0|->, the system is degenerate.
{|+>,|->} are the eigenstates of Sz. .

At t=0 the system is in the state |->, but the Hamiltonian is H0+W=H, with

.

In the {|+>,|->} the matrix of W is

.

The matrix of H is

.

The eigenvalues of H are , and the corresponding eigenfunctions are

.

In this problem  .

.

.

.

If then .

(b)  If  with n an odd integer, then .

The Hamiltonian of a system of two spin ½ 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.

• Solution

(a)  The first term (A) is the part of the Hamiltonian that does not depend on spin, the second term is the spin-spin interaction, and the third term is the interaction of the spins with an external magnetic field.  In the {|S,Sz>} basis the matrices of are diagonal.

,

.

, .

The matrix of H is therefore diagonal and the matrix of S2 is diagonal.

, .

Since all these matrices are diagonal, they all commute with each other.

(b)  The eigenvalues of H are the diagonal matrix elements.  The eigenvectors are the corresponding basis vectors, and  .