Write the Schrödinger equation (3.22) if $\mathit{\psi }$is a function ofx, and $\mathit{V}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathit{\omega }}^{\mathbf{2}}{\mathit{x}}^{\mathbf{2}}$ (this is a one-dimensional harmonic oscillator). Find the solutions ${\mathit{\psi }}_{\mathbf{n}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and the energy eigenvalues ${\mathit{E}}_{\mathbf{n}}$ . Hints: In Chapter 12, equation (22.1) and the first equation in (22.11), replace xby $\mathit{\alpha }\mathit{x}$where $\mathit{\alpha }\mathbf{=}\sqrt{\mathbf{m}\mathbf{\omega }\mathbf{/}\mathbf{\hslash }}$. (Don't forget appropriate factors of $\mathit{\alpha }$for the ${\mathit{x}}^{\mathbf{\text{'}}}$ 's in the denominators of $\mathit{D}\mathbf{=}\raisebox{1ex}{$\mathbf{d}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}\mathbf{x}$}\right.$and ${\mathit{\psi }}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{=}\raisebox{1ex}{${\mathbf{d}}^{\mathbf{2}}\mathbf{\psi }$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}{\mathbf{x}}^{\mathbf{2}}$}\right.$.) Compare your results for equation (22.1) with the Schrödinger equation you wrote above to see that they are identical if ${\mathit{E}}_{\mathbf{n}}\mathbf{=}\mathbf{(}\mathbf{n}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{)}\mathit{\hslash }\mathit{\omega }$. Write the solutions ${\mathit{\psi }}_{\mathbf{n}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$of the Schrödinger equation using Chapter 12, equations (22.11) and (22.12).

The Hermite Differential equation is given:

$y_n^{\text{'}\text{'}}-x^2{y}_n=-(2n+1)y_n$

A linear partial differential equation that governs the wave function of a quantum-mechanical system is known as Schrödinger equation.

Rewrite Hermite differential equation in the form of one dimensional harmonic oscillator.

Replace xby$\alpha x$.

Here,$\alpha =\sqrt{\frac{m\omega }{\hslash }}$.

The modified linear operator is:

$\begin{array}{c}D=\frac{d}{dx}\\ D^{\text{'}}=\sqrt{\frac{\hslash }{m\omega }}D\end{array}$

Use the modified linear Differential Operator to rewrite the Hermit Differential Equation.

$\begin{array}{c}(D^{\text{'}}+\sqrt{\frac{m\omega }{\hslash }}x)y_n=(\sqrt{\frac{\hslash }{m\omega }}D+\mathrm{sqrt}\frac{m\omega }{\hslash }x)y_n\\ =(\sqrt{\frac{\hslash }{m\omega }}{y}_n^{\text{'}}+\sqrt{\frac{m\omega }{\hslash }}x{y}_n)\end{array}$

$\begin{array}{c}(D^{\text{'}}-\sqrt{\frac{m\omega }{\hslash }}x)(\sqrt{\frac{\hslash }{m\omega }}{y}_n^{\text{'}}+\sqrt{\frac{m\omega }{\hslash }}x{y}_n)=(\sqrt{\frac{\hslash }{m\omega }}D-\sqrt{\frac{m\omega }{\hslash }}x)(\sqrt{\frac{\hslash }{m\omega }}{y}_n^{\text{'}}+\sqrt{\frac{m\omega }{\hslash }}x{y}_n)\\ =\frac{\hslash }{m\omega }{y}_n^{\text{'}\text{'}}-\frac{m\omega }{\hslash }{x}^2{y}_n+y_n\end{array}$

For a one-dimensional harmonic oscillator, generate the Schrödinger equation.

$\begin{array}{l}\frac{\hslash }{m\omega }{y}_n^{\text{'}\text{'}}-\frac{m\omega }{\hslash }{x}^2{y}_n+y_n=-2n{y}_n\\ -\frac{{\hslash }^2}{2m}{y}_n^{\text{'}\text{'}}+\frac{1}{2}m{\omega }^2{x}^2{y}_n-\frac{\hslash \omega }{2}{y}_n=\hslash \omega n{y}_n\\ -\frac{{\hslash }^2}{2m}{y}_n^{\text{'}\text{'}}+\frac{1}{2}m{\omega }^2{x}^2{y}_n=\frac{\hslash \omega }{2}{y}_n+\hslash \omega n{y}_n\\ -\frac{{\hslash }^2}{2m}{y}_n^{\text{'}\text{'}}+\frac{1}{2}m{\omega }^2{x}^2{y}_n=\hslash \omega (n+\frac{1}{2})y_n\end{array}$

Hence the solution is:

$\begin{array}{c}{y}_n\left(x\right)={\psi }_n\left(x\right)\\ ={(\sqrt{\frac{\hslash }{m\omega }}D-\sqrt{\frac{m\omega }{\hslash }}x)}^n{e}^{-\frac{m\omega }{2\hslash }{x}^2}\end{array}$