The Pauli Exclusion Principle — Question 4

Question 4

Sharing one spatial orbital forces the spin singlet

Two electrons occupy the same normalized spatial orbital u. Their total wave function must change sign under exchange. Use $|\uparrow\rangle$ and $|\downarrow\rangle$ as orthonormal spin states.

1. Construct the normalized total state.

2. Find the possible total-spin quantum number and the probabilities of the two opposite z outcomes.

Question 4 - Solution

Sharing one spatial orbital forces the spin singlet

(a) Method and calculation

$$\psi_{\mathrm{space}}=u(1)u(2)\quad\text{is symmetric under exchange}.$$$$\boxed{\Psi=u(1)u(2)\frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt2}}.$$

(b) Interpretation and further calculation

$$\boxed{S=0,\quad M_S=0}.$$$$\boxed{P(\uparrow\downarrow)=P(\downarrow\uparrow)=1/2,\quad P(\uparrow\uparrow)=P(\downarrow\downarrow)=0}.$$

Physical check

The singlet is antisymmetric in spin, making the total state antisymmetric. A parallel-spin pair cannot occupy one identical spatial orbital.