In Exercise 5.19, an unexpected negative sign arose in a rotation by 2π about the z-axis. But perhaps this was merely due to some unwise choice on our part. The purpose of this problem is to suggest...

In Exercise 5.19, an unexpected negative sign arose in a rotation by 2π about the z-axis. But perhaps this was merely due to some unwise choice on our part. The purpose of this problem is to suggest that, on the contrary, negative signs inevitably arise in rotation of spin-1/2 systems.

(a) Suppose the operator R describes a rotation by π/2 about the y-axis. We might hope that such an operator would change the spin states like so:

with no negative signs. Given the relationship between
 (Eq. 3.14), show that no linear operator R could act in this way.

(b) Suppose instead that the operator R acts by

with various phase factors α, β, and γ . (We have chosen the overall phase of R to eliminate any phase factor in the first rotation.) Find all of the phase factors and show that R⁴