Question

Is there a simple derivation for the probability of measuring the same spin state for a particle at an angle, theta, to its spin axis as cos(theta/2) squared if the spin direction is already known?

Answer 1

Suppose the Stern Gerlach apparatus points along z-axis and we prepare a spin up state ie, . Now, we rotate the apparatus so that it lies along a arbitary direction . I will derive for a case where lies in the x-z plane (ny = 0). What is the probability of getting spin up state?

Let us denote the component of along by

where are the pauli matrices

Now we can answer the above question.

The probabbility of getting same state (ie ) is

I have given simple derivation by assuming ny = 0. A more general case where points in an arbitary direction in space can be derived but the mathematics is rigorous.