QuantumIsHard
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Homework Statement
The magnetic moment for an electron is [itex]\mu<sub>e</sub> = -e/m S<sub>e</sub>[/itex].
The magnetic moment for a positron is [itex]\mu<sub>e</sub> = +e/m S<sub>e</sub>[/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<sub>0</sub>[/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<sub>J</sub> = 1 + {j*(j+1) – l*(l+1) + 3/4}/{2*j*(j+1)}[/itex]
Energy shift:
[itex]E<sub>Z</sub><sup>1</sup> = \mu<sub>B</sub> g<sub>J</sub> B<sub>ext</sub> m<sub>J</sub>[/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 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 [itex]\mu<sub>B</sub>, g<sub>J</sub>, and, m<sub>J</sub>[/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.