Principles of Quantum Mechanics · Problem 14.3.1

Spin — R. Shankar

Physics — Quantum Mechanics · spin-spinor-eigenket

Problem

Let us verify the above corollary explicitly. Take some spinor with components

\alpha = \rho_1 e^{i\phi_1}

and

\beta = \rho_2 e^{i\phi_2}

. From

\langle \chi | \chi \rangle = 1

, deduce that we can write

\rho_1 = \cos(\theta/2)

and

\rho_2 = \sin(\theta/2)

for some

\theta

. Next pull out a common phase factor so that the spinor takes the form in Eq. (14.3.28a). This verifies the corollary and also fixes

\hat{n}