QM M04 2
From QED
The Hamiltonian is:
Which we can rewrite using the dot product of the spin operator:
Particle 1 has spin S, particle 2 has spin 1/2, so we will get:
The total spin :
The energy (thus taking on 2S + 2 states), while , so it takes on 2S states.
Since we have two spin 1/2 particles, we can express every state as the up or down of each of the two particles:
We apply the hamiltonian (and use our knowledge of Pauli Matrices to apply Sx and Sy to Sz eigenstates):
And:
Next:
Lastly:
So the eigenvalue , while the eigenvalue .
We use the same eigenvectors as they are all eigenvectors of Sz. Finding eigenvalues:
Similarly:
For :
For :
If we only consider these two states, then:
Which gives us the determinant:
Which has solutions . I will not find the normalized eigenstates because it is not fun.
This page was recovered in October 2009 from the Plasmagicians page on Prelim_M04_QM2 dated 14:05, 24 April 2006.