What Is the Total Wavefunction of a 2-Electron System with 1s and 2p States?

[jumphigh]

Homework Statement

Ignoring the repulsion force between two electrons, one of the electrons is in 1s state and the other is in 2p state. what is the total wavefunction of the system that is made up of the multiplication of the spatial wavefunctions and spin values?

Homework Equations

so, if electron(1) is in 1s state and e(2) is in 2p state, their spatial wavefunction will be :
u₁(r₁)u₂(r₂) - u₁(r₂)u₂(r₁)
with spins of
chi_-(1) chi₊2 and so on

The Attempt at a Solution

there are 8 possible products of spatial wavefcn and their spins for both electrons individually, I think there are only 4 combinations that are allowed by exclusion principle, those are:

{u₁(r₁)u₂(r₂) - u₁(r₂)u₂(r₁) }χ₊(1) χ₊(2)

{u₁(r₁)u₂(r₂) - u₁(r₂)u₂(r₁) }χ_-(1) χ_-(2)

u₁(r₁)u₂(r₂) {χ_-(1) χ₊(2) } - u₁(r₂)u₂(r₁) {χ_-(2) χ₊(1) }

u₁(r₁)u₂(r₂) {χ_-(2) χ₊(1) } - u₁(r₂)u₂(r₁) {χ_-(1) χ₊(2) }

Im not sure if the last combination is necessary, it kinda looks like the third one, any ideas? does the whole combination makes sense?

Thanks

 

[mark726]

for your question! It seems like you have the right idea with the possible combinations of the spatial wavefunctions and spin values for the two electrons. The exclusion principle does indeed limit the allowed combinations, so the first two combinations you listed are the only ones that are allowed. The last two combinations are not necessary and are actually just equivalent to the first two combinations. This is because the spin values of the two electrons are interchangeable, so the spin values can be swapped without changing the overall wavefunction. So, the total wavefunction for the system would be:

{u1(r1)u2(r2) - u1(r2)u2(r1) }χ+(1) χ+(2)

OR

{u1(r1)u2(r2) - u1(r2)u2(r1) }χ-(1) χ-(2)

I hope this helps clarify things for you! Keep up the good work in your studies as a scientist.