The Pauli Exclusion Principle — Question 10

Question 10

The one-electron density of a determinant retains both occupied orbitals

Two parallel-spin electrons occupy orthonormal spatial orbitals u and v in $\Psi=[u(1)v(2)-v(1)u(2)]/\sqrt2$. Define number density $n(\boldsymbol r)=2\int|\Psi(\boldsymbol r,\boldsymbol r_2)|^2d^3r_2$.

1. Derive n and its normalization.

2. For a region A with $\int_A|u|^2d^3r=.7$ and $\int_A|v|^2d^3r=.2$, find the expected number of electrons in A and the probability for a randomly selected electron to be in A.

Question 10 - Solution

The one-electron density of a determinant retains both occupied orbitals

(a) Method and calculation

$$\int u^*(2)v(2)d^3r_2=0\Rightarrow\boxed{n(\boldsymbol r)=|u(\boldsymbol r)|^2+|v(\boldsymbol r)|^2}.$$$$\boxed{\int n(\boldsymbol r)d^3r=1+1=2}.$$

(b) Interpretation and further calculation

$$\boxed{\langle N_A\rangle=\int_A n\,d^3r=.7+.2=.9}.$$$$\boxed{P_A^{\mathrm{random}}=\langle N_A\rangle/2=.45}.$$

Physical check

An expected number is not itself the probability that at least one electron lies in A. That event requires the joint density and its correlations.