Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Possible values and their Probability of Measuring S^2 - Spin

  1. Aug 27, 2010 #1
    1. The problem statement, all variables and given/known data
    I have a two spin 1/2 particles. The Hamiltonian for the system is given as H = w1 S1z + w2 S2z. I need to find the possible values and their probabilities when I measure S^2 at some later time T. Also the Initial state \Psi (0) = a | [tex]\uparrow[/tex] [tex]\downarrow[/tex] > + b | [tex]\downarrow[/tex] [tex]\uparrow[/tex]>


    2. Relevant equations



    3. The attempt at a solution

    Now I know for a 2 spin 1/2 particle system, s = 1 and 0.

    The eigenvalue equation for S2 is S2|sm> = hbar2 ( s ( s+1) )|sm>

    So the possible values are 2 \hbar^2 and 0.

    I know at some later time, the state will look like \Psi(t) = a e{-iE_1 t/ \hbar} + b e{-iE_2 t/ \hbar}

    and I can find E_1 and E_2

    However, how do i find the probabilities?

    If I was just looking for S_z probabilities, I know it would be a2 for spin up and b2 for spin down. I also know that if I was looking for S_x I would need to evolve the coefficients in time. However, how do I measure the probabilities of S2?
     
  2. jcsd
  3. Aug 27, 2010 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You need to express [itex]|\psi(t)\rangle[/itex] in terms of the eigenstates of S2. Do you know how to do that?
     
  4. Aug 29, 2010 #3
    Hmm, I'm not positive... is that writing it in |10> and |00> states? Im not exactly sure how to do this. Can you help get me moving in the right direction?

    thanks for the help.
     
  5. Aug 29, 2010 #4

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Yes, that's what I mean. Do you know how to express those states as linear combinations of [itex]|\uparrow\,\downarrow\,\rangle[/itex] and [itex]|\downarrow\,\uparrow\,\rangle[/itex]? If not, you should figure out how to do that. It's probably covered in your textbook as it's a pretty common example of the addition of angular momentum.
     
  6. Aug 31, 2010 #5
    Ok, I think I got it,

    so using:

    e1 = [TEX]\frac{1}{\sqrt{2}}[/TEX](|+,-> + |-,+>) with eigenvalue 2hbar^2

    and

    e2 = [TEX]\frac{1}{\sqrt{2}}[/TEX](|+,-> - |-,+>) with eigenvalue 0

    I rewrite: |+,-> as (e1 + e2)*(sqrt(2)/2) and |-,+> as (e1 - e2)*(sqrt(2)/2)

    Combine e1 and e2 terms. And then the coefficients squared give me the probabilities of measuring each eigenvalue as a function of time.

    Sound good?

    Thanks for your help.
     
  7. Aug 31, 2010 #6

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Yup, good job!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook