- #1

QuantumIsHard

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## Homework Statement

The magnetic moment for an electron is [itex]\mu

_{e}= -e/m S

_{e}[/itex].

The magnetic moment for a positron is [itex]\mu

_{e}= +e/m S

_{e}[/itex].

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 [itex]H

_{0}[/itex] 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.

## Homework Equations

g-factor:

[itex]g

_{J}= 1 + {j*(j+1) – l*(l+1) + 3/4}/{2*j*(j+1)}[/itex]

Energy shift:

[itex]E

_{Z}

^{1}= \mu

_{B}g

_{J}B

_{ext}m

_{J}[/itex]

## 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) [itex]H0 = -\hbar^2/{2m} * ( {\delta^2}/{\delta^2 r1} ) - \hbar^2/{2m} * ( {\delta^2}/{\delta^2 r2} )[/itex] and [itex]H' = {k*e^2}/{|r1-r2|^2}[/itex], 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 m

_{j}, I believe I could do this with a Clebsh-Gordan table.

e) I think I could just use the above equation once I know [itex]\mu

_{B}, g

_{J}, and, m

_{J}[/itex]

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.