Question

Write all the possible wave functions for two electrons in two generic orbitals Phi_1, and Phi_2.

Answer 1

ANS:

For a 2- electron system, 2 characteristics should be considered:

1. electron's coordinates in wavefunctions must be such that the electrons are indistinguishable , i.e,

              | $\small \psi$ ²| = $\small \psi$ . $\small \psi$ as every electron is identical.

2.This requirement is according to pauli's exclusion principle. It is that the wavefunction for a 2 electron system must change sign any time the coordinates of any 2 electrons are permuted.

     $\small \psi$ (1,2....i,..j...,N) = - $\small \psi$ (1,2,...j,...i,....N)

Considering the above two requirements, the wavefunction for 2 electrons must be antisymmetric w.r.t interchange of spins and indistinguishable.

Thus, the possible wavefunctions are:

¹ $\small \psi$ = 1/ $\small \sqrt{2$ ( $\small \phi$ _a(1) $\small \phi$ _b(2) + $\small \phi$ _b(1) $\small \phi$ _a(2)) * 1/ $\small \sqrt{2}$ ( $\small \alpha 1\beta 2 - \beta 1\alpha 2$ )              M_s=0

³ $\small \psi$ =1/ $\small \sqrt{2}$ ( $\small \phi$ _a(1) $\small \phi$ _b(2) - $\small \phi$ _b(1) $\small \phi$ _a(2)) * $\tiny \begin{Bmatrix} \alpha 1\alpha 2 & & \\ 1/\sqrt{2}(\alpha 1\beta 2+\beta 1\alpha 2)& & \\ \beta 1\beta 2 & & \end{Bmatrix}$               M_s =1, 0, -1