Quantum Mechanics - Ground State of Helium Atom

[Tangent87]

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.

Can anyone help me please?

 

[fzero]

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.

 

[Tangent87]

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

 

[TriKri]

(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.

 

[paranormal]

The Pauli exclusion principle is a fundamental principle in quantum mechanics that states that no two identical fermions (particles with half-integer spin) can occupy the same quantum state simultaneously. This means that in the ground state of the helium atom, the two electrons must have different quantum numbers such as spin, energy level, and orbital angular momentum.

In terms of the wavefunction, the Pauli exclusion principle results in an antisymmetric wavefunction for the ground state of helium. This means that if the positions of the two electrons are swapped, the wavefunction changes sign. This is due to the fact that electrons are indistinguishable and cannot be labeled as "electron 1" and "electron 2."

Your confusion in case (b) is due to a misunderstanding of the antisymmetric nature of the wavefunction. While it is true that the antisymmetric expression gives zero when the two electrons are in the same quantum state, this does not mean that the ground state wavefunction of the helium atom is zero. The wavefunction is a complex function that describes the probability of finding the electrons in certain positions and states. It can have non-zero values even if the antisymmetric expression gives zero.

In case (c), the Wikipedia article is referring to the total wavefunction of the helium atom, which is a combination of the spatial wavefunction and the spin wavefunction. The spatial wavefunction is symmetric, meaning that it does not change sign when the positions of the electrons are swapped. However, the spin wavefunction is antisymmetric, resulting in an overall antisymmetric wavefunction for the helium atom.

In summary, the Pauli exclusion principle has important consequences for the ground state of the helium atom. It results in an antisymmetric wavefunction, which reflects the indistinguishability of the two electrons and their requirement to occupy different quantum states.