Question

Two-electron system. (A) Which of the following are physically valid wavefunctions for a two-electron system such as the He atom or the H2 molecule? (B) What property makes them physically valic wavefunctions? (a) y(12)-1s(1)a(1) . 1s(2)A(2) (c) Ψ(1,2)-Isa(1)IsE(2)-1sa(2)1sE(1). Multi-electron atom. Consider a sodium cation, Nat. (A) How many variables does the multi-electron wavefunction describing its ground state depend on? (B) How many kinetic-energy, repulsion and attraction terms are in the associated Hamiltonian?

Answer 1

Electrons have half integral spins therefore they are considered as fermions. And for fermions , total wavefunction have to be anti symmetry. ( Pauli’s principle)

$ ( considering it as psi here )

condition for symmetric wavefuncion is $(1,2)= $(2,1)

Condition for antisymmtric is $(1,2) = -$(2,1).

In above given first wavefunction, alpha(1) beta(2) on spin interchanging transforms into alpha(2)beta(1). And it does not satisfy the condition of antisymmetric wavefunction.

In second wavefunction, alpha(1)beta(2) + alpha(2)beta(1) on spin interchanging became alpha(2)beta(1) + alpha(1)beta(2). And here it satisfy the condition of symmetric  wavefuction

In third wave function, alpha(1)beta(2)-alpha(2)beta(1) on spin interchanging became alpha(2)beta(1)-alpha(1)beta(2).

alpha(1)beta(2)-alpha(2)beta(1) = -(alpha(2)beta(1)-alpha(1)beta(2))

this satisfies the condition of antisymmetric wavefunction to be physically valid for fermions.