- #1
Aleolomorfo
- 73
- 4
Homework Statement
Considering the atom made of an electron and a positron. The spin-orbit Hamiltonian is:
$$H=\frac{e^2}{4\nu^2c^2r^34\pi\epsilon_0}\vec{L}\cdot\vec{S}$$
with ##\vec{L}## the relative angular momentum, with ##\vec{S}## the total spin and ##\mu## the reduced mass. Finding the levels in which ##3d## states are split.
Homework Equations
The Attempt at a Solution
First I show you the way I solved the problem.
$$\vec{J}=\vec{L}+\vec{S}$$
$$\vec{S}\cdot\vec{L} = \frac{J^2-L^2-S^2}{2}$$
So I can rewrite the Hamiltonian:
$$H=A(J^2-L^2-S^2)$$
with ##A## which is equal to al the constants. The splitting in energy is:
$$\Delta E = A\hbar^2 (J(J+1) - \frac{21}{4})$$
In the last step I put ##S=\frac{1}{2}## and ##L=2##. Then I calculate the possible ##J##'s:
$$|L-S|\le J \le L+S$$
So there is a double splitting with ##J=\frac{3}{2}## and ##J=\frac{5}{2}##.
I think the reasoning is right (isn't it?), but I have a question about the spin. I put ##S=\frac{1}{2}## automatically as in the hydrogen spin-orbit coupling. But why do I consider only the spin of the electron and not the total spin of the system, so the spin of the electron and the spin of the positron?