A Modern Approach to Quantum Mechanics · Problem 5.8

Wave Mechanics in One Dimension — John S. Townsend

Physics — Quantum Mechanics · spin-correlation-singlet-state

Problem

Measurements of the spin components along two arbitrary directions

\mathbf{a}

and

\mathbf{b}

are performed on two spin-

\frac{1}{2}

particles in the singlet state

|0, 0\rangle

. The results of each measurement are denoted by

\pm 1

, depending on whether the measurement finds the particle spin-up or spin-down along that particular axis. Denoting by

P_{\pm\pm}(\mathbf{a}, \mathbf{b})

the probabilities of obtaining

\pm 1

along

\mathbf{a}

for particle 1 and

\pm 1

along

\mathbf{b}

for particle 2, the average value for the product of spins is given by

E(\mathbf{a}, \mathbf{b}) = P_{++}(\mathbf{a}, \mathbf{b}) + P_{--}(\mathbf{a}, \mathbf{b}) - P_{+-}(\mathbf{a}, \mathbf{b}) - P_{-+}(\mathbf{a}, \mathbf{b})

Show that

E(\mathbf{a}, \mathbf{b}) = -\cos\theta_{ab}

where

\theta_{ab}

is the angle between

\mathbf{a}

and

\mathbf{b}

. If

\mathbf{a}

and

\mathbf{b}

are unit vectors, then we can write

E(\mathbf{a}, \mathbf{b}) = -\mathbf{a} \cdot \mathbf{b}