Quantum Background

In an excited state of the helium atom, one electron is in the 1s orbital and the other in the 2s orbital. Since electrons are fermions, the total wavefunction must be antisymmetric under exchange of the two electrons.

The total wavefunction is a product of:

• Spatial (orbital) part
• Spin part

In the singlet state, the spin part is antisymmetric, so the spatial part must be symmetric.

Spin Part (Antisymmetric)

For the singlet state, the spin wavefunction is:

χsinglet = (1/√2) [α(1)β(2) - β(1)α(2)]

This form is antisymmetric under exchange of the two electrons.

Spatial Part (Symmetric)

The spatial wavefunction for one electron in 1s and the other in 2s is written as:

ψspace = (1/√2) [ψ1s(1)ψ2s(2) + ψ2s(1)ψ1s(2)]

This is symmetric under exchange.

Total Spin-Orbital Wavefunction (Singlet)

The total wavefunction is the product of the symmetric spatial and antisymmetric spin parts:

Ψ = (1/√2) [ψ1s(1)ψ2s(2) + ψ2s(1)ψ1s(2)] × (1/√2) [α(1)β(2) - β(1)α(2)]

Simplified:
Ψ = (1/2) [ψ1s(1)ψ2s(2) + ψ2s(1)ψ1s(2)] × [α(1)β(2) - β(1)α(2)]

This total wavefunction is properly antisymmetric under particle exchange, satisfying the Pauli Exclusion Principle.