A spin-half particle is prepared in the state

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Spin one-half, SU(2) and two-level dynamics

Problem

|\psi\rangle = \cos(\pi/8)\,|\uparrow\rangle + \sin(\pi/8)\,|\downarrow\rangle

, where

|\uparrow\rangle

and

|\downarrow\rangle

are the eigenstates of

\hat\sigma_z

with eigenvalues

+1

and

-1

. ((a)) Give the unit vector

\hat n = (n_x,n_y,n_z)

that this state occupies on the Bloch sphere. ((b)) Find the probability that a measurement of spin along the

x

-axis gives

+\hbar/2