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Magnetic Moment- Spin System - Probability

  1. Mar 6, 2013 #1

    VVS

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    1. The problem statement, all variables and given/known data

    Spin-system:
    A spin-system has N = 30 localized particles with spin quantum number s = ½ and magnetic moment µ0. m = M/µ0 is the magnetic moment M in vertical direction (ms = +½) in units of µ0.
    Calculate the probability PN(m) for the total magnetic moment M to achieve in vertical direction the value m
    a) Without magnetic field (p = q =1/2) and
    b) With magnetic field (p = 0,7; q = 0,3),
    under the assumption that p und q = 1-p are the probabilities for ms = +1/2 and ms = -1/2, respectively.
    Draw the values for PN(m) in both cases.


    2. Relevant equations

    all given in 1.

    3. The attempt at a solution

    View attachment Problem_Sheet_1_Exercise_3.pdf

    I just don't know how the spin is related to the magnetic moment.
    Is m=1/2 or -1/2 or M=1/2 or -1/2
     
  2. jcsd
  3. Mar 7, 2013 #2
    Hint: What is the formula for "spin angular momentum" ?
    What is the formula for "magnetic moment" ?
    Once you know the two, you can see the obvious relation.
     
  4. Mar 7, 2013 #3

    VVS

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    Hi,

    Thanks for the hint. I checked online, but I can't find any formulae for sping angular momentum and magnetic moment.

    Could you please give me another hint?

    thank you
     
  5. Mar 8, 2013 #4
    Spin angular momentum = [h√{S*(S+1)}]/2π

    Magnetic moment = √{4S*(S+1)}

    Now can you carry out further ?

    S is spin quantum number.
     
  6. Mar 8, 2013 #5

    VVS

    User Avatar

    Thanks,

    From those two formulae it is clear that Spin Angular Momentum = h/(4*pi)*magnetic moment.

    But I am not sure how this is supposed to help me with the problem. The sentence I didn't understand was:

    m = M/µ0 is the magnetic moment M in vertical direction (ms = +½) in units of µ0. Calculate the probability PN(m) for the total magnetic moment M to achieve in vertical direction the value m under the assumption that p und q = 1-p are the probabilities for ms = +1/2 and ms = -1/2, respectively.

    Is M=±1/2 or M/μ0 = ±1/2?
     
  7. Mar 8, 2013 #6
    I cannot understand this:

    µ0 is magnetic moment of 30 localized particles, fine. But what is m?
    I guess that M being vertical component, we have µ0 sinθ=M. This means m=sinθ. What is ms? In my textbook, ms is spin quantum number, but here already its denoted by S. Might be that you should contact other HHs.
     
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