Question

Show that the symmetric and anti symmetric combinations of $\mathbf{(}\mathbf{8}\mathbf{-}\mathbf{19}\mathbf{)}$and$\mathbf{(}\mathbf{8}\mathbf{-}\mathbf{20}\mathbf{)}$are solutions of the two. Particle Schrödinger equation$\mathbf{(}\mathbf{8}\mathbf{-}\mathbf{13}\mathbf{)}$of the same energy as${\mathbf{\psi }}_{\mathbf{n}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}{\mathbf{\psi }}_{\mathbf{m}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}$, the unsymmetrized product$\mathbf{(}\mathbf{8}\mathbf{-}\mathbf{17}\mathbf{)}$.

Short answer

It is proved that symmetric and asymmetric combination are the solution of two particles Schrodinger equation.

Step-by-step solution

Given information

The symmetric wave function is${\mathrm{\psi }}_{\text{s}}({\mathrm{x}}_1,{\mathrm{x}}_2)={\mathrm{\psi }}_{\text{n}}\left({\mathrm{x}}_1\right){\mathrm{\psi }}_{\text{n}}\left({\mathrm{x}}_2\right)+{\mathrm{\psi }}_{\text{n}}\left({\mathrm{x}}_1\right){\mathrm{\psi }}_{\text{n}}\left({\mathrm{x}}_2\right)$ .

The asymmetric wave function is${\mathrm{\psi }}_{\text{A}}({\mathrm{x}}_1,{\mathrm{x}}_2)={\mathrm{\psi }}_{\text{n}}\left({\mathrm{x}}_1\right){\mathrm{\psi }}_{\text{m}}\left({\mathrm{x}}_2\right)-{\mathrm{\psi }}_{\text{m}}\left({\mathrm{x}}_1\right){\mathrm{\psi }}_{\text{n}}\left({\mathrm{x}}_2\right)$ .

Concept of complex number:

The expression for combination of symmetric and asymmetric wave function is given by ${\mathbf{\psi }}_{\mathbf{\text{SA}}}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{)}\mathbf{=}{\mathbf{\psi }}_{\mathbf{\text{n}}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}{\mathbf{\psi }}_{\mathbf{\text{n}}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}\mathbf{\pm }{\mathbf{\psi }}_{\mathbf{\text{n}}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}{\mathbf{\psi }}_{\mathbf{\text{n}}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}$

Evaluate symmetric and asymmetric wave function

The expression for combination of symmetric and asymmetric wave function is calculated as,

Apply Schrodinger wave equation

By Schrodinger wave equation in one dimension,

Considering equation (1) and (2)

$\left\{\begin{array}{c}-\frac{{\mathrm{h}}^2}{2\mathrm{m}}\left[\left(\frac{{\partial }^2}{\partial {\mathrm{x}}_1^2}\right)+\left(\frac{{\partial }^2}{\partial {\mathrm{x}}_2^2}\right)\right]{\mathrm{\psi }}_{\text{SA}}({\mathrm{x}}_1,{\mathrm{x}}_2)\\ +[\mathrm{U}\left({\mathrm{x}}_1\right)+\mathrm{U}\left({\mathrm{x}}_2\right)]{\mathrm{\psi }}_{\text{SA}}({\mathrm{x}}_1,{\mathrm{x}}_2)\end{array}\right\}={\mathrm{E\psi \psi }}_{\text{SA}}({\mathrm{x}}_1,{\mathrm{x}}_2)$

Thus, the symmetric and asymmetric combination obeys the Schrodinger equation with two particle solution.