Introduction to Quantum Mechanics · Problem 4.66

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · spin_uncertainty

Problem

Deduce the condition for minimum uncertainty in

S_x

and

S_y

(that is, equality in the expression

\sigma_{S_x}\sigma_{S_y} \geq (\hbar/2)|\langle S_z\rangle|

), for a particle of spin 1/2 in the generic state (Equation 4.139). Answer: With no loss of generality we can pick

a

to be real; then the condition for minimum uncertainty is that

b

is either pure real or else pure imaginary.