- #1

unscientific

- 1,734

- 13

## Homework Statement

(a) Find the spatial wavefunction

(b)Show anti-symmetric wavefunctions have larger mean spacing

(c) Discuss the importance of this

(d)Determine the total orbital angular momentum

(e)Hence find the ground state term for Z=15[/B]

## Homework Equations

## The Attempt at a Solution

__Part(a)__[/B]

The overall wavefunction must be anti-symmetric due to Pauli's exclusion principle. Since the spin can either be singlet (anti-symmetric) or tripplet (symmetric), the spatial part must be written as a symmetric and anti-symmetric combination of ## u_{A(r_1)} u_{B(r_2)}## and ##u_{A(r_2)}u_{B(r_1)} ##.

__Part(b)__

Overlap leads to terms like:

[tex]\phi \phi^* = \frac{1}{2} \left[ | u_{A(r(1)} |^2 |u_{B(r_1)}|^2 + | u_{A(r(2)} |^2 |u_{B(r_2)}|^2 \pm 2 Re \left( u_{A(r_1)}u_{B(r_2)} u^*_{B(r_1)}u^*_{A(r_2)} \right) \right] [/tex]

Hence when spin is aligned (symmetric), the spatial part must be anti-symmetric.

Don't we get ## \phi_{AS} \phi_{AS}^* < \phi_{S} \phi_{S}^* ##?

Which is strange, as I know that spatially anti-symmetric wavefunctions are further away.

__Part(c)__

Spatially symmetric -> electrons closer -> more shielding -> higher energy (Para-helium)

spatially anti-symmetric -> electrons further -> less shielding -> lower energy (Ortho-helium)

__Part (d)__

Due to spin-orbit coupling, won't the total angular momentum ##L = l_1 + l_2 + l_3 = 3##?

__Part (e)__

The ground term is simply ## ^4P_3##.