Question

Determine whether each wavefunctions are valid? Which follows the Pauli Exclusion Rule.

b. [ls(1)2s(2) - 1s(2)2s(1)]a(1)a(2) [1s(1)2s(2)-1s(2)2s(1)][αφβ(2)-a(2)β(1)] [1s(1)2s(2)a(1)β(2)]-[1s(2)2s(1)a(2)β(1)] + [1s(1)2s(2)α(2)βφ] [1 s(2)2s(1 )α(1) β(2)] c. d.

Answer 1

a.

$\psi = 1s(1)2s(2)[\alpha(1)\beta(2)-\alpha(2)\beta(1)]$

Verdict: This wavefunction is NOT VALID.

Reason: Since the electrons are identical and indintiguishable, we must use a linear combination for the spatial part of the wave function.

Pauli's exclusion principle states that particle with half integer spin(electron has spin $\pm 1/2$ ) must have an antisymmetric wavefunction(with respect to change of coordiantes, here interchange a and b to check )

In this example, even though the wavefunction is invalid, it obeys Pauli's exclusion principle as it has a symmetric spatial part and antisymmetric spin part making the total wavefunction antisymmetric.

b.

$\psi = [1s(1)2s(2)-1s(2)2s(1)]\alpha(1)\alpha(2)$

$\psi_{spatial} = [1s(1)2s(2)-1s(2)2s(1)] \ (anti-symmetric)$

$\psi_{spin} = \alpha(1)\alpha(2) (symmetric)$

Hence, the total wavefunction is anti-symmetric.

It also does not violate the indintinguishability of the electrons with a linear combination as spatial part.

Hence, it is a VALID wavefunction which obeys Pauli's Exclusion Principle.

c.

$\psi = [1s(1)2s(2)-1s(2)2s(1)][\alpha(1)\beta(2)-\alpha(2)\beta(1)]$

$\psi_{spatial} = [1s(1)2s(2)-1s(2)2s(1)] \ (anti-symmetric)$

$\psi_{spin} = [\alpha(1)\beta(2)-\alpha(2)\beta(1)] (anti-symmetric)$

Since both the spatial and spin part are ant-symmetric, the total wavefunction is symmetric which is not allowed under Pauli's exclusion principle.

Hence, the wave fucntion does not obey Pauli's exclusion principle and is INVALID.

d.

$\psi = [1s(1)2s(2)\alpha(1)\beta(2)] - [1s(2)2s(1)\alpha(2)\beta(1)] + [1s(1)2s(2)\alpha(2)\beta(1)]-[1s(2)] 2s(1)\alpha(1)\beta(2)]$

Grouping the common terms

$\psi = 1s(1)2s(2)[\alpha(1)\beta(2)+\alpha(2)\beta(1)] - 1s(2)2s(1)[\alpha(2)\beta(1)+\alpha(1)\beta(2)] \\ \Rightarrow \psi = [1s(1)2s(2)-1s(2)2s(1)][\alpha(1)\beta(2)+\alpha(2)\beta(1)]$

Now separating the spatial part and spin part:

$\psi_{spatial} = [1s(1)2s(2)-1s(2)2s(1)] (anti-symmetric)$

$\psi_{spin} = [\alpha(1)\beta(2)+\alpha(2)\beta(1)] (symmetric)$

Since, one part is symmetric and another is antisymmetric. This wavefunction obeys Pauli's exclusion principle and is a VALID wavefunction.