An Interlude — Question 1

Question 1

Spin projections distinguish state counting from vector magnitude

A massive particle has spin quantum number s=3/2. Use $S_z=m_s\hbar$ and $|\mathbf S|=\sqrt{s(s+1)}\hbar$.

1. List allowed spin projections and find the number of states.

2. Compare maximum projection with spin magnitude and classify its spin statistics.

Question 1 - Solution

Spin projections distinguish state counting from vector magnitude

(a) Method and calculation

$$\boxed{m_s=-3/2,-1/2,+1/2,+3/2,\quad g=2s+1=4}.$$$$\boxed{S_{z,\max}=3\hbar/2}.$$

(b) Interpretation and further calculation

$$\boxed{|\mathbf S|=\sqrt{15}\hbar/2>S_{z,\max}}.$$$$s\in\mathbb Z+\tfrac12\Rightarrow\boxed{\text{fermion; identical particles obey Fermi statistics}}.$$

Physical check

Spin magnitude is not one of its projection eigenvalues. Spatial and other internal quantum numbers also enter the full single-particle state.