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The Hermite Differential equation is given:

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

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

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

Replace xby$伪x$.

Here,$伪=\sqrt{\frac{m蝇}{\mathrm{鈩�}}}$.

The modified linear operator is:

$\begin{array}{c}D=\frac{d}{dx}\\ D^{\text{'}}=\sqrt{\frac{\mathrm{鈩�}}{m蝇}}D\end{array}$

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

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

$\begin{array}{c}(D^{\text{'}}鈭�\sqrt{\frac{m蝇}{\mathrm{鈩�}}}x)(\sqrt{\frac{\mathrm{鈩�}}{m蝇}}{y}_n^{\text{'}}+\sqrt{\frac{m蝇}{\mathrm{鈩�}}}x{y}_n)=(\sqrt{\frac{\mathrm{鈩�}}{m蝇}}D鈭�\sqrt{\frac{m蝇}{\mathrm{鈩�}}}x)(\sqrt{\frac{\mathrm{鈩�}}{m蝇}}{y}_n^{\text{'}}+\sqrt{\frac{m蝇}{\mathrm{鈩�}}}x{y}_n)\\ =\frac{\mathrm{鈩�}}{m蝇}{y}_n^{\text{'}\text{'}}鈭�\frac{m蝇}{\mathrm{鈩�}}{x}^2{y}_n+y_n\end{array}$

For a one-dimensional harmonic oscillator, generate the Schr枚dinger equation.

$\begin{array}{l}\frac{\mathrm{鈩�}}{m蝇}{y}_n^{\text{'}\text{'}}鈭�\frac{m蝇}{\mathrm{鈩�}}{x}^2{y}_n+y_n=鈭�2n{y}_n\\ 鈭�\frac{{\mathrm{鈩�}}^2}{2m}{y}_n^{\text{'}\text{'}}+\frac{1}{2}m{蝇}^2{x}^2{y}_n鈭�\frac{\mathrm{鈩�}蝇}{2}{y}_n=\mathrm{鈩�}蝇n{y}_n\\ 鈭�\frac{{\mathrm{鈩�}}^2}{2m}{y}_n^{\text{'}\text{'}}+\frac{1}{2}m{蝇}^2{x}^2{y}_n=\frac{\mathrm{鈩�}蝇}{2}{y}_n+\mathrm{鈩�}蝇n{y}_n\\ 鈭�\frac{{\mathrm{鈩�}}^2}{2m}{y}_n^{\text{'}\text{'}}+\frac{1}{2}m{蝇}^2{x}^2{y}_n=\mathrm{鈩�}蝇(n+\frac{1}{2})y_n\end{array}$

Hence the solution is:

$\begin{array}{c}{y}_n\left(x\right)={蠄}_n\left(x\right)\\ ={(\sqrt{\frac{\mathrm{鈩�}}{m蝇}}D鈭�\sqrt{\frac{m蝇}{\mathrm{鈩�}}}x)}^n{e}^{鈭�\frac{m蝇}{2\mathrm{鈩�}}{x}^2}\end{array}$