# Zeeman Shift on Positronium

1. Feb 23, 2013

### QuantumIsHard

1. The problem statement, all variables and given/known data

The magnetic moment for an electron is $\mue = -e/m Se$.
The magnetic moment for a positron is $\mue = +e/m Se$.
In the ground state, the quantum numbers are n=1 and l=0.

a) What is the physical mechanism for the Zeeman shift?
b) Write the Hamiltonian and identify $H0$ and H'.
c) In the weak-field limit, the two spins couple together to make a total spin F. What are the possible values for F in the ground state of positronium?
d) Continuing in the weak field limit, the Hamiltonian needs to be written in terms of the total spin F. Project each of the spins S onto the total spin F and find the value of the Lande g-factor for each value of F.
e) Sketch the energy level shift as a function of applied B for each value of F in the weak field limit.

2. Relevant equations

g-factor:
$gJ = 1 + {j*(j+1) – l*(l+1) + 3/4}/{2*j*(j+1)}$

Energy shift:
$EZ1 = \muB gJ Bext mJ$

3. The attempt at a solution

a) The motion of the electron and positron will produce a magnetic field experienced by the other. The Zeeman shift will factor in this field.

b) $H0 = -\hbar^2/{2m} * ( {\delta^2}/{\delta^2 r1} ) - \hbar^2/{2m} * ( {\delta^2}/{\delta^2 r2} )$ and $H' = {k*e^2}/{|r1-r2|^2}$, with H just being the sum of the two.

c) F=1 when the spins align and F=0 when the spins are opposite.

d) If I knew j and mj, I believe I could do this with a Clebsh-Gordan table.

e) I think I could just use the above equation once I know $\muB, gJ, and, mJ$

I'm nowhere near confident with (a)-(c) and am stuck entirely on (d) and (e). Any help would be greatly appreciated.

This is my first post on this forum, so my apologies for any formatting issues.

2. Feb 24, 2013

### andrien

3. Feb 24, 2013

### PhysicsGente

You are explaining hyperfine splitting not the Zeeman effect. The Zeeman effect is the effect on the atom's magnetic moment due to an external magnetic field.

For part b, you can write the hamiltonian as $H=\mu\cdot B_{ext}.$ (think about why).
I don't know what you are referring to by H_0 and H'.

Last edited: Feb 24, 2013
4. Feb 25, 2013

### andrien

I have given a reference in post#3 there,which deals with both.