QM M04 2
From QED
The Hamiltonian is:

Which we can rewrite using the dot product of the spin operator:
![\mathcal{H}=\frac{a}{2}\left[J^{2}-S_{1}^{2}-S_{2}^{2}\right]](../../../images/math/9/4/5/94530bbb2fa784aac7fd80dce39165bb.png)
Particle 1 has spin S, particle 2 has spin 1/2, so we will get:
![E=\frac{a}{2}\left[j(j+1)\hbar^{2}-S(S+1)\hbar^{2}-\frac{3\hbar^{2}}{4}\right]](../../../images/math/3/6/e/36ed1e548c7a2b945a36a18c91f90af2.png)
The total spin :
![E=\frac{a\hbar^{2}}{2}\left[\left(S\pm\frac{1}{2}\right)\left(S\pm\frac{1}{2}+1\right)-S(S+1)-\frac{3}{4}\right]](../../../images/math/b/e/7/be7999846374fb73fe1398ec871a6446.png)
![E=\frac{a\hbar^{2}}{2}\left[S^{2}\pm\frac{S}{2}+S\pm\frac{S}{2}+\frac{1}{4}\pm\frac{1}{2}-S^{2}-S-\frac{3}{4}\right]=\frac{a\hbar^{2}}{2}\left[\pm S\pm\frac{1}{2}-\frac{1}{2}\right]](../../../images/math/0/a/1/0a1a496935a48d7731d7ddd4d7a62971.png)
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.