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Find the energy levels of a spin s = 3/2 particle

  1. Nov 22, 2015 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant 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}##

    3. 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)
     
  2. jcsd
  3. Nov 22, 2015 #2

    blue_leaf77

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    Science Advisor
    Homework Helper

    Those relations hold only for spin one-half particles. You don't need matrix representation actually, just use the fact that ##S^2 = S_x^2 + S_y^2 + S_z^2## to modify the first three terms contained in the bracket in the Hamiltonian.
     
  4. Nov 22, 2015 #3
    Hi Blue leaf, I realised this after I made the post and I think I have the correct answer now.

    I just have one question: Is there a convention as to which order you evaluate the different ##m_j## values when computing the matrix elements? Is it ascending or descending?
     
  5. Nov 22, 2015 #4

    DrClaude

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    Staff: Mentor

    It is usually descending, just like for a spin-1/2 particle:
    $$\hat{S}_z=\frac{1}{2}\hbar\begin{bmatrix}1&0\\0&-1\end{bmatrix}$$

    spin-3/2:
    $$\hat{S}_z=\frac{1}{2}\hbar\begin{bmatrix}3&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-3 \end{bmatrix}$$
    But it is always better to specify which convention is used.
     
  6. Nov 23, 2015 #5
    Ok thanks for the help!
     
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