Question

Show that both the symmetric wave function $\psi_S$ and the antisymmetric wave function $\psi_A$ of Eqs. (4.12) and (4.13), respectively, will yield probability density functions that are not in any way affected by the labelling of the two electrons. Only clue from the book is below: We write a symmetric wave function $\psi_S$ for the two-electron system as $\psi_S = \frac{1}{\sqrt{2}}[\psi_{T_1} + \psi_{T_2}]$ (4.12) and an antisymmetric wave function $\psi_A$ for the two-electron system as $\psi_A = \frac{1}{\sqrt{2}}[\psi_{T_1} - \psi_{T_2}]$ (4.13) If $\psi_S$ is used in place of $\psi_T$ to calculate the probability density function $|\psi_S|^2$, the result will be independent of the choice of the subscripts. In addition, since both $\psi_{T_1}$ and $\psi_{T_2}$ are valid solutions to Schrödinger's equation (Eq. (4.8)) and since $\psi_S$ is a linear combination of these solutions, it follows that $\psi_S$ is also a valid solution. The same argument applies to $\psi_A$ (see Problem 4.20).

          Show that both the symmetric wave function $\psi_S$ and the antisymmetric wave function $\psi_A$ of
Eqs. (4.12) and (4.13), respectively, will yield probability density functions that are not in any
way affected by the labelling of the two electrons.
Only clue from the book is below:
We write a symmetric wave function $\psi_S$ for the two-electron system as
$\psi_S = \frac{1}{\sqrt{2}}[\psi_{T_1} + \psi_{T_2}]$ (4.12)
and an antisymmetric wave function $\psi_A$ for the two-electron system as
$\psi_A = \frac{1}{\sqrt{2}}[\psi_{T_1} - \psi_{T_2}]$ (4.13)
If $\psi_S$ is used in place of $\psi_T$ to calculate the probability density function $|\psi_S|^2$, the result
will be independent of the choice of the subscripts. In addition, since both $\psi_{T_1}$ and $\psi_{T_2}$ are
valid solutions to Schrödinger's equation (Eq. (4.8)) and since $\psi_S$ is a linear combination
of these solutions, it follows that $\psi_S$ is also a valid solution. The same argument applies to
$\psi_A$ (see Problem 4.20).

Show that both the symmetric wave function and the antisymmetric wave function of
Eqs. (4.12) and (4.13), respectively, will yield probability density functions that are not in any
way affected by the labelling of the two electrons.
Only clue from the book is below:
We write a symmetric wave function for the two-electron system as
= (1)/(√(2))[ψT1 + ψT2] (4.12)
and an antisymmetric wave function for the two-electron system as
= (1)/(√(2))[ψT1 - ψT2] (4.13)
If is used in place of to calculate the probability density function ||^2, the result
will be independent of the choice of the subscripts. In addition, since both ψT1 and ψT2 are
valid solutions to Schrödinger's equation (Eq. (4.8)) and since is a linear combination
of these solutions, it follows that is also a valid solution. The same argument applies to
(see Problem 4.20).

Added by Thomas A.

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