The Pauli Exclusion Principle — Question 5

Question 5

Parallel spins require an antisymmetric spatial wave function

Two electrons have the symmetric spin state $|\uparrow\uparrow\rangle$ and occupy distinct orthonormal spatial orbitals u and v. Neglect interactions.

1. Construct their normalized spatial state.

2. Evaluate its amplitude at equal positions and explain the resulting same-spin coincidence suppression.

Question 5 - Solution

Parallel spins require an antisymmetric spatial wave function

(a) Method and calculation

$$\Psi_{\mathrm{space}}(1,2)=C[u(1)v(2)-v(1)u(2)].$$$$\boxed{\Psi_{\mathrm{space}}=\frac{u(1)v(2)-v(1)u(2)}{\sqrt2}}.$$

(b) Interpretation and further calculation

$$\Psi_{\mathrm{space}}(\boldsymbol r,\boldsymbol r)=\frac{u(\boldsymbol r)v(\boldsymbol r)-v(\boldsymbol r)u(\boldsymbol r)}{\sqrt2}=0.$$$$\boxed{|\Psi_{\mathrm{space}}(\boldsymbol r,\boldsymbol r)|^2=0}.$$

Physical check

This zero follows from exchange antisymmetry even with no repulsive force. Opposite-spin electrons are not subject to the same spatial coincidence zero.