Rotating Spin

A spin-1/2 particle interacts with a magnetic field B = B₀ẑ through the Pauli interaction H = μσ . B, where μ is the magnetic moment and σ = (σ_x, σ_y, σ_z) are the Pauli spin matrices. At t = 0 a measurement determines that the spin is pointing along the positive x-axis What is the probability that it will be pointing along the negative y-axis at a later time t?

SOLUTION

Let us quantize the spin states along the z-axis so that spin up and spin down are denoted by

(1)

(2)

(3)

(4)

(5)

The eigenstates of σ_x are ѱ_x for pointing along the +x-axis, and for the –x-axis

(1)

(7)

(8)

(9)

At time t = 0 we start in state ѱ_x. Later this state becomes

(10)

The amplitude for pointing in the negative y-direction is found by taking the matrix element with  The probability is the square of the absolute magnitude of this amplitude:

(11)

(12)