Electron Spin — Question 1

Question 1

Spin magnitude cannot be identified with its largest projection

An electron has $s=1/2$. Use $S^2=s(s+1)\hbar^2$ and $S_z=m_s\hbar$. A measurement prepares $m_s=+1/2$.

1. Find the spin magnitude and its z projection.

2. Find $\langle S_x^2+S_y^2\rangle$ and explain why the spin cannot have all its angular momentum along z.

Question 1 - Solution

Spin magnitude cannot be identified with its largest projection

(a) Method and calculation

$$|S|=\sqrt{s(s+1)}\hbar=\boxed{\sqrt3\hbar/2}.$$$$\boxed{S_z=\hbar/2},\qquad S_z/|S|=1/\sqrt3.$$

(b) Interpretation and further calculation

$$\langle S_x^2+S_y^2\rangle=s(s+1)\hbar^2-m_s^2\hbar^2.$$$$\boxed{\langle S_x^2+S_y^2\rangle=\hbar^2/2\ne0}.$$

Physical check

The projection ratio is not evidence for a literal classical spinning axis. Transverse spin components fluctuate in a z eigenstate.