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Homework Help: Spin 1/2 in B-field polarization direction?

  1. Apr 27, 2007 #1
    spin 1/2 in B-field polarization direction?!?!

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

    A spin 1/2 particle with magnetic momnet mu=mnot*sigma is placed in

    uniform B-field, where the magnetic field is pointing in (theta,phi) direction

    in spherical coordinates

    At t=0, the particle is in the (1 0) state, i.e. sigma(z)=1 and the polarization

    direction is =e3 (z hat direction).

    Find out the polarization direction of the particle at t>0.



    2. Relevant equations

    the theory of spin is like angular momentum

    magnetic moment mu=-gamma*S

    S=spin matrix

    S^2|sm>=hbar^2s(s+1)|sm>

    and Sz|sm>=hbar*m|sm>


    3. The attempt at a solution

    Not sure how to approach this one, i assume that the particle is at rest in the B-field

    the Hamiltonian for a spinning charged particle in B-field is H=-gamma*(B dot S)

    this is a time-DEPENDENT perturbation

    how can i realte what is given to the polarization direction?

    cheers
    nate:biggrin:
     
  2. jcsd
  3. Apr 28, 2007 #2
    ok, let me try this again

    H=-mu*B (this is the interaction energy between spin and B-field)

    for a particle of spin 1/2, with magnetic moment mu=mu(s)*S

    the Hamiltonian is h=1/2*m(p-q/c*A)^2-mu*B

    B=BX+BY+BZ (where i need to put B into spherical coordinates)

    so if i have the B-field in the z-direction, A=(-By,0,0)

    so....

    H=1/2*m[(px^2+py^2+pz^2)+(2*q*B*px/c)*y+(qB/c)^2*y^2]-(mu*B/S)Sz

    is this correct? i have a Hamilton with Sz in it which is given as sigma(z)=1 (pauli spin matrix)
     
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