# Quantum Mechanics - Ground State of Helium Atom

I have confused myself with this by reading a combination of Wikipedia, books and my QM notes and I'm afraid I need someone to untangle me please.

Basically what I want to know is, what are the consequences of the Pauli exclusion principle
for the ground state of the helium atom?

Here's my confusion: (a). The Pauli exclusion principle says that two electrons (fermions) must occupy a totally antisymmetric state, thus the ground state wavefunction of the helium atom must be antisymmetric.

(b). However the electrons are identical particles and thus the Pauli exclusion principle says that the antisymmetric expression gives zero. Therefore the the ground state wavefunction of the helium atom must be zero.

(c). Also, the wikipedia article on the Helium atom (http://en.wikipedia.org/wiki/Helium_atom) seems to suggest that the ground state wavefunction of the helium atom must be symmetric (unless I'm misreading it which could well be the case).

I know case (b) is wrong because a zero wavefunction is not normalizable but I can't see the flaw in my logic.

fzero
Homework Helper
Gold Member
Electrons have intrinsic spin $$\pm 1/2$$. This can be included as a factor in the electron wavefunction: $$\Psi(\vec{r},m) = \psi(\vec{r})\chi_m$$. When constructing multiple particle wavefunctions, we can symmetrize over the spatial and spin parts of the wavefunction independently. The spin-statistics theorem requires that the overall wavefunction be antisymmetric. This could be accomplished either by having the spatial part be symmetric and the spin part be antisymmetric (opposite spins for He) or vice versa. The proper treatment of all interactions in the Hamiltonian determines which configuration is the ground state.

Ah okay thanks, so the overall wavefunction must be antisymmetric but we have to have one of the spatial or spin parts being symmetric.

(a). The Pauli exclusion principle says that two electrons (fermions) must occupy a totally antisymmetric state

No, that is the definition of a fermion.

This 'asymmetry' however doesn't say anything about the wave function of a single fermion, since it includes a swap between two identical fermions which you don't have in the case with the helium atom.

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