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

1. Apr 27, 2007

### valtorEN

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

2. Apr 28, 2007

### valtorEN

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)