Confusing statement about the Pauli exclusion principle

Question

While reading "Francis A. Carey, Richard J. Sundberg - Advanced Organic Chemistry Part A. Structure and Mechanisms-Springer (2007)", I came across the following:

The paragraph says that there are two major reasons for electron pairs repelling each other.

1. Electrostatic repulsion
2. Pauli exclusion principle

Now, the statement says that, in accordance with the Pauli exclusion principle, only two electron can occupy the same point in space and that those two electrons must have opposite spins. However, as far as I know, that is not the statement of Pauli exclusion principle, or is it equivalent to the following statement?

No two electrons can have the same four quantum numbers.

This is a tricky question, since we really shouldn't be talking about electrons occupying a single point in space (given the uncertainty principle). Also, wouldn't electrostatic repulsion be enough to assert that no two electrons can occupy the same point in space regardless of the spin?

Answer 1

Although this is not a common way to represent the Pauli exclusion principle, it is indeed one consequence of it. Specifically, we start with the statement that a system of identical electrons must be antisymmetric with respect to any exchange of the electrons, which is the actual postulate that Pauli proposed.[The derivation of this bit is quite complicated and comes from relativistic quantum field theory.] That is, $$\psi(q_1, q_2, . . . q_n)=-\psi(q_2, q_1,. . . . q_n)$$ where each "$q_i$" is an electron described by a given set of quantum numbers.

Although we don't consider electrons to be localized at specific points in space, we can consider the value of their wavefunctions as a function of three spatial coordinates and a spin coordinate, i.e. $x,y,z$ and $m_s$ or $\theta, \phi, r$ and $m_s$. If these are equal for $q_1$ and $q_2$ (ie two particles have the same spatial location and spin), we would have that $$\psi(q_1, q_1, . . . q_n)=-\psi(q_1, q_1,. . . . q_n)$$ which is only true if $\psi =0$, which means that the wave function associated with this set of electrons has zero probability of being in a state where two electrons of the same spin also have the same spatial coordinates.

There's a more complete description of this in Levine's Quantum chemistry book in chapter 10.

UPDATE based on comments: Since $\psi$ is a continous function, the fact that it has zero amplitude when two electrons have the same spatial and spin values also means that the amplitude approaches zero as two electrons get closer together, which is a way of saying that the probability (amplitude squared) of the system existing in a state where two electrons are close together (but not in the same position) is low and approaches zero as the electrons get closer together (because it necessarily equals zero for the state where they are at the same point). We describe this result as "repulsion" of the electrons from one another.