Transmission coefficient of spin-1/2 in magnetic field

[the_ron]

Homework Statement

A particle of mass m, (nonrelativistic) energy E, spin 1/2 and magnetic moment $\mu*\sigma$ is traveling in the positive x-direction from large negative x. For x > 0, there is a constant magnetic field B in the positive z-direction. There is no field for x < 0.

a) Find the transmission coefficient as a function of energy E for spin up in the B-direction, and also for spin down in the B-direction.

b) Suppose now the spin is initially in the x-direction. Describe how the spin of the transmitted particle varies, if at all. Include in your discussion the case of very low incoming kinetic energy.

Homework Equations

The usual 1-d potential plane wave states for a transmitted.
For x < 0 Psi(x) = e^ikx + r*e^-ikx
For x > 0 Psi(x) = t*e^iKx

Where k = Sqrt[2mE]/hbar and K = Sqrt[2m(E-V)]/hbar

The Attempt at a Solution

I think I understand part A correctly. The transmission coefficient, t, splits into two possible values, depending on whether there's spin up or spin down.

For part b, the wave function on x>0 will be a superposition of the two eigenstates in part a.
I've written it as Psi(x) = 1/Sqrt[2] * (t⁺*e^iK⁺x + t^-*e^iK^-x),
where t,k^+,- are the two values for the Sx spin state.

I've went through and determined the values of t⁺ and t^- by using the continuity of the wave function and its derivative at the origin, and by renormalizing the state to 1, which doesn't seem correct now...

The question also asks to consider very low kinetic energy, which I'm not sure how to handle. In this case, maybe only the spin up component of the wave function will be transmitted.

I'm not looking for help towards an exact solution. I'm just curious as to whether I've handled the second part of the problem generally correctly.

Thanks!

 

[diazona]

You can't normalize plane waves, so don't bother with that step... other than that, it sounds like you're on the right track. (renormalization is something entirely different, by the way)

For the low-energy case, look at your results for the two cases, spin up and spin down, and think about why one of them might not make sense if E is smaller than some value.