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Solving coupled differential equations for spin-1/2 in a B-field

  1. Nov 13, 2012 #1
    1. The problem statement, all variables and given/known data

    (Sorry, I don't know how to use latex)

    Solve the TDSE for a spin half nucleus in a B-field where B_z = B0, B_y= B1 cos(ωt) and B_x = B1 sin(ωt).
    Use vector and matrix representation. You will get coupled differential equations for a and b, look for solutions of the form a = a_0 ω_a e^(i ω_a t) and b = b_0 ω_b e^(i ω_b t)


    2. Relevant equations

    |ψ> = column vector (a,b)


    3. The attempt at a solution


    I've done all of the problem but have got stuck trying to find the solutions in the form a = a_0 ω_a e^(i ω_a t) and b = b_0 ω_b e^(i ω_b t)

    I have (da/dt) = (-i [itex]\gamma[/itex] t /2) (B_0 a + B_1 b e^(-i [itex]\omega [/itex] t) )

    (db/dt) = (i [itex]\gamma[/itex] t / 2 ) (B_0 b + B_1 a e^(i [itex]\omega[/itex] t ) )

    but don't know how to separate them. My lecturer gave me the hint that the equations have to hold at all times, not just time = t, so I tried using t=0, getting rid of the exponentials, but that didn't help
     
  2. jcsd
  3. Nov 14, 2012 #2
    these eqn can be made algebraic by using just the differential of a(t) and b(t) which you have just written.
     
  4. Nov 14, 2012 #3
    Can yo ugive me a hint at how to start with that though? I have no idea what to do
     
  5. Nov 15, 2012 #4
    you can put value of a(t)=a0e(iωat) and similarly for b(t) in the differential eqn. and you can see after that there is only algebraic eqn to solve.you can put t=o after to get those a0 and others.It seems you are doing something wrong so check out your calculation.You can see feynman lectures vol. 3 for similar problems although solved.
     
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