A solved problem for spins

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This problem is added by team 8; Source: Introduction to Quantum Mechanics,D. Griffiths,Problem 4-34.

Problem: An electron is at rest in an oscillating magnetic field

where and are constants.

(a) Construct the Hamiltonian matrix for this system.

(b) The electron starts out (at t = 0) in the spin-up state with respect to the x-axis [that is,]. Determine at any subsequent time. Beware.' This is a time-dependent Hamiltonian, so you cannot get in the usual way from stationary states. Fortunately, in this case you can solve the time-dependent Schr/Sdinger equation directly.

(c) Find the probability of getting if you measure

(d) What is the minimum field required to force a complete flip in ?

Solution:

(a)

(b)

with

, so


(c)

(d)

The argument of must reach (so P=1) , or