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

[HBarker]

Homework Statement

(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)

Homework Equations

|ψ> = column vector (a,b)

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 $\gamma$ t /2) (B_0 a + B_1 b e^(-i $\omega$ t) )

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

 

[andrien]

these eqn can be made algebraic by using just the differential of a(t) and b(t) which you have just written.

 

[HBarker]

Can yo ugive me a hint at how to start with that though? I have no idea what to do

 

[andrien]

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.