How Long Must Neutrons Stay in a Magnetic Field to Reverse Spin Direction?

[v_pino]

Homework Statement

A beam of neutrons traveling in the x direction initially has all its spins up along the z direction. The neutrons pass through a region in which there is a uniform 1 tesla magnetic field in the y direction. What is the minimum time t the neutrons need to remain in the field to exit it with their spins along -z? The magnetic moment of the neutron is u=-0.966x10^-26 J/T.

Homework Equations

Is this question to do with time-dependent Hamiltonian? If so, does this equation help?

$$B=B_0+B_1=B_0 \hat z + b_1(\hat x cos \omega t - \hat y sin \omega t)$$

The Attempt at a Solution

Please help me get started. Thank you.

 

[ardie]

the energy required to flip each neutron is H = u . B, where the terms on the right are vectors. in a time varying field B, and using the Heisenberg picture in which the states are constants of time (which is very useful for solving this problem) one can define the time evolution operator such that the generator is the hamiltonian for the system. there's quite a lot involved in answering this question which I may or may not be able to help you with. ultimately the time evolution operator gives you the new states when operating on the old states, and there you can see the coefficients which give you the equation for resonance times of the required flipping.
almost any book on advanced quantum mechanics will help you with this, as its one of the most fundamental calculations