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pondzo
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Homework Statement
Find the energy levels of a spin ##s=\frac{3}{2}## particle whose Hamiltonian is given by:
##\hat{H}=\frac{a_1}{\hbar^2}(\hat{S}^2-\hat{S}_x^2-\hat{S}_y^2)-\frac{a_2}{\hbar}\hat{S}_z## where ##a_1## and ##a_2## are constants.
Homework Equations
In the ##\hat{S}_z## basis ##\hat{S}_z## and ##\hat{S}^2## have the following matrix representations:
##\hat{S}_z=\frac{1}{2}\hbar\begin{bmatrix}1&&0\\0&&-1\end{bmatrix}##
##\hat{S}^2=\frac{3}{4}\hbar^2\begin{bmatrix}1&&0\\0&&1\end{bmatrix}##
The Attempt at a Solution
We can rewrite the Hamiltonian as follows:
##\hat{H}=\frac{a_1}{\hbar^2}(2\hat{S}_z^2-\hat{S}^2)-\frac{a_2}{\hbar}\hat{S}_z##
Subbing the matrices in the "relevant equations" I get the following matrix representation for the Hamiltonian:
##\hat{H}=\frac{-1}{4}\begin{bmatrix}a_1+2a_2&&0\\0&&a_1-2a_2\end{bmatrix}##
I'm not sure where to go from here... since ##s=\frac{3}{2}## then ##m_s=-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}##. Is the next step to work out the energy for each of these ##m_s## values?
So find: ##\langle\frac{3}{2},-\frac{3}{2}|\hat{H}|\frac{3}{2},-\frac{3}{2}\rangle,\langle\frac{3}{2},-\frac{1}{2}|\hat{H}|\frac{3}{2},-\frac{1}{2}\rangle,\langle\frac{3}{2},\frac{1}{2}|\hat{H}|\frac{3}{2},\frac{1}{2}\rangle,\langle\frac{3}{2},\frac{3}{2}|\hat{H}|\frac{3}{2},\frac{3}{2}\rangle,##? If so, I am a little confused as to the vector forms of some of those bras and kets.. Help is appreciated!
(Oh and by the way, my Lecturer made a mistake when he said a particle with spin s = 3/2, but he said to do the question regardless)