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Time development of a spin state

  1. Jun 24, 2013 #1
    1. The problem statement, all variables and given/known data
    For a homework problem, I have to work out ##\frac{d(\langle \hat{S}_x \rangle )}{dt}## for the four-by-four spin matrix ##\hat{S}_x##. I have a spin matrix ##\hat{S}_x## and I need to use the generalized Ehrenfest theorem. My problem is that I'm not sure whether my approach is mathematically valid (see below) as it envolves derivatives and matrices in the same expression.


    2. Relevant equations

    ##\frac{d(\langle \hat{S}_x \rangle )}{dt}=\langle \left [ \hat{S}_x , \hat{H} \right ] \rangle##

    3. The attempt at a solution

    I'm starting off with the commutation relation on the RHS, but I run into immediate difficulties:

    $$\left [ \hat{S}_x , \hat{H} \right ] = \hat{S}_x \left ( -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\right ) - \left ( -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\right )\hat{S}_x$$

    As the spin matrix is a four-by-four matrix, I'm not sure how to proceed from here (or even if what I've got is correct). I considered letting the operators act on a function which I would define as a column vector with a functional dependence on x, but I don't know how to perform the differentiation on a column vector.
    There must be a simpler path to take...can anyone help?
    P.S. I have a time-dependente value for the expectation of ##\hat{S}_x##, which I could simply differentiate. The question specifically tells me to use the Generalized Ehrenfest theorem though.
     
  2. jcsd
  3. Jun 25, 2013 #2
    Well first of all, where are you getting your equation from, because from when I learned it and in every quantum mechanics book Ive ever seen, you're missing a factor of [tex]\frac{1}{i\hbar}[/tex] in front of the right hand side you have given us. Maybe youre just leaving out constant though, which is fine if youre just trying to figure out the math!

    Also is there more information you are withholding? What makes me say this is because you state that you have a time dependent value for [tex]\hat{S} _x[/tex], so where is this coming from? I need a little more information to try and deduce how you want to solve this problem.
     
  4. Jun 26, 2013 #3
    You're quite right, I missed the constant out accidentally, as I was focussing too much on the commutation relation that I couldn't figure out.
    I was given the initial spin state of a single particle moving in a magnetic field and Larmor's frequency. I've since realised that I can put the Hamiltonian in a matrix form using this information, and that makes the commutation relation a lot easier to figure out. Providing I have got this right, I assume it should be just a question of taking the expectation value using the sandwich integral and then multiply by the constant that I'd missed out to begin with...
     
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