A spin-1/2 particle is prepared in the state | = 1 2 (| + e^ i | ), = /3

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch The wave function is not the state

Problem

A spin-1/2 particle is prepared in the state

|\psi\rangle = \frac{1}{\sqrt2}\big(|{\uparrow}\rangle + e^{i\varphi}|{\downarrow}\rangle\big),\qquad \varphi = \pi/3,

where

|{\uparrow}\rangle,|{\downarrow}\rangle

are the eigenstates of

\hat S_z

with eigenvalues

\pm\hbar/2

. A Stern-Gerlach magnet is oriented along

x

, with eigenstate

|{+x}\rangle = \frac{1}{\sqrt2}(|{\uparrow}\rangle+|{\downarrow}\rangle)

for outcome

S_x=+\hbar/2

. ((a)) Someone argues that since "all phase is unobservable", the probability of measuring

S_x = +\hbar/2

must be the same

1/2

regardless of

\varphi

. Compute

P(S_x=+\hbar/2)

correctly for

\varphi=\pi/3

. ((b)) State the general formula

P(S_x=+\hbar/2)

as a function of

\varphi

and check that it gives the expected probabilities at

\varphi=0

and

\varphi=\pi