What Are the Wave Functions for Singlet and Triplet States of H2?

[loserlearner]

Homework Statement

The wave function for a system of two hydrogen atoms can be described approximately in
terms of hydrogen wave functions.
(a) Give the complete wave functions for the lowest states of the system for singlet and triplet
spin configurations. Sketch the spatial part of each wave function along a line through the
two atoms.
(b) Sketch the effective potential energy for the atoms in the two cases as functions of the
internuclear separation. Neglect rotation of the system. Explain the physical origin of the
main features of the curves, and of any differences between them.

Homework Equations

The complete wavefunction is ψ(x₁,x₂)\chi(s₁,s₂) for a two particle system, assuming the spatial wavefunction and the spin are separable.

Hydrogen ground state is ψ₁₀₀= \frac{1}{\sqrt{\pi}}(\frac{1}{a})^{3/2}e^{-\frac{r}{a}}

Singlet (anti-symmetric) state is | 0 0 > = \frac{1}{\sqrt{2}}(\uparrow\downarrow-\downarrow\uparrow)
Triplet (symetric) states are:
| 1 1 > = \uparrow\uparrow
| 1 0 > = \frac{1}{\sqrt{2}}(\uparrow\downarrow+\downarrow\uparrow)
| 1 -1 > = \downarrow\downarrow

The Attempt at a Solution

a)
I try to compose the total wavefunction with the spatial part and the spin states, but I am not sure how to deal with the fact that there are two electrons that are under the Coulomb potential of both protons.

Since there are two electrons in this system both spin-1/2, would the system be viewed as spin-1 and therefore a symmetric total wavefunction?

b)
I am clueless as to what the question is asking. Maybe with an understanding of part a), it would become clear.

I would really appreciate it if someone can point me in the right direction. Thanks!

 

[DrClaude]

a)
I try to compose the total wavefunction with the spatial part and the spin states, but I am not sure how to deal with the fact that there are two electrons that are under the Coulomb potential of both protons.

That's not important, as you are only approximating the exact wave function. What is important to keep in mind is that you are combining two wave functions that are centered on different atoms.

Since there are two electrons in this system both spin-1/2, would the system be viewed as spin-1 and therefore a symmetric total wavefunction?

No no no! The Pauli principle applies to many particle wave functions. The wave function must be antisymmetric under the exchange of two identical fermions.