91影视

• A differential equation that describes matter in quantum mechanics in terms of the wave-like properties of particles in a field.
• Its answer is related to a particle's probability density in space and time.

$$\begin{gathered} \frac{\mathrm{d}}{\mathrm{dt}}{鈭�}_{-鈭�}^{鈭�}{{\mathrm{唯}}_1}^{*}{\mathrm{唯}}_2\mathrm{dx}={鈭�}_{-鈭�}^{鈭�}\frac{鈭�}{鈭�t}\left({{\mathrm{唯}}_1}^{*}{\mathrm{唯}}_2\right)dx \\ ={鈭�}_{-鈭�}^{鈭�}\left({\mathrm{唯}}_2\frac{鈭�{\mathrm{唯}}_1^{*}}{鈭�\mathrm{t}}+{{\mathrm{唯}}^{*}}_1\frac{鈭�{\mathrm{唯}}_2}{鈭�\mathrm{t}}\right)dx \end{gathered}$$

But, from Schrodinger equation

$\frac{鈭�{\mathrm{唯}}_2}{鈭�\mathrm{t}}=\frac{i\sout{h}}{2m}\frac{{鈭�}^2{\mathrm{唯}}_2}{鈭�x^2}-\frac{i}{\sout{h}}V{唯}_2$

And

$\frac{鈭�{\mathrm{唯}}_1^{*}}{鈭�\mathrm{t}}=-\frac{i\sout{h}}{2m}\frac{{鈭�}^2{\mathrm{唯}}_2}{鈭�x^2}-\frac{i}{\sout{h}}V{唯}_1^{*}$

Thus,

localid="1658552483464" $$\begin{gathered} \frac{d}{dt}{鈭�}_{-鈭�}^{鈭�}{{唯}_{\mathit{1}}}^{\mathit{*}}{唯}_{\mathit{2}}dx={\mathit{鈭�}}_{\mathit{-}\mathit{鈭�}}^{\mathit{鈭�}}\mathit{(}{\mathit{唯}}_{\mathit{2}}\left[-\frac{i\sout{h}}{2m}\frac{{鈭�}^2{唯}_1^{*}}{鈭�x^2}+\frac{i}{\sout{h}}\right]\mathit{+}{\mathit{唯}}_{\mathit{1}}^{\mathit{*}}\mathit{[}\mathit{-}\frac{\mathit{i}\sout{\mathit{h}}}{\mathit{2}\mathit{m}}\frac{{\mathit{鈭�}}^{\mathit{2}}{\mathit{唯}}_{\mathit{2}}}{\mathit{鈭�}{\mathit{x}}^{\mathit{2}}}\mathit{+}\frac{\mathit{i}}{\sout{\mathit{h}}}\mathit{]}\mathit{V}{\mathit{唯}}_{\mathit{2}}\mathit{)}dx \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}=-\frac{i\sout{h}}{\mathit{2}m}{鈭�}_{-鈭�}^{鈭�}\left({\mathrm{唯}}_{\mathit{2}}\frac{{\mathit{鈭�}}^{\mathit{2}}{\mathrm{唯}}_{\mathit{1}}^{\mathit{*}}}{\mathit{鈭�}{\mathrm{x}}^{\mathit{2}}}-{\mathrm{唯}}_{\mathit{1}}^{\mathit{*}}\frac{{\mathit{鈭�}}^{\mathit{2}}{\mathrm{唯}}_{\mathit{2}}}{\mathit{鈭�}{\mathrm{x}}^{\mathit{2}}}\right)\mathrm{dx} \\ =-\frac{i\sout{h}}{\mathit{2}m}{鈭�}_{-鈭�}^{鈭�}\left({\mathrm{唯}}_{\mathit{2}}\frac{{\mathit{鈭�}}^{\mathit{2}}{\mathrm{唯}}_{\mathit{1}}^{\mathit{*}}}{\mathit{鈭�}{\mathrm{x}}^{\mathit{2}}}-{\mathrm{唯}}_{\mathit{1}}^{\mathit{*}}\frac{{\mathit{鈭�}}^{\mathit{2}}{\mathrm{唯}}_{\mathit{2}}}{\mathit{鈭�}{\mathrm{x}}^{\mathit{2}}}\right)\mathrm{dx} \\ =\mathit{-}\frac{i\sout{h}}{\mathit{2}m}{鈭�}_{-鈭�}^{鈭�}\frac{鈭�}{鈭�\mathrm{x}}\left({\mathrm{唯}}_{\mathit{2}}\frac{\mathit{鈭�}{\mathrm{唯}}_{\mathit{1}}^{\mathit{*}}}{\mathit{鈭�}\mathrm{x}}-{\mathrm{唯}}_{\mathit{1}}^{\mathit{*}}\frac{\mathit{鈭�}{\mathrm{唯}}_{\mathit{2}}}{\mathit{鈭�}\mathrm{x}}\right)\mathrm{dx} \\ \frac{\mathrm{d}}{\mathrm{dt}}{鈭�}_{-鈭�}^{鈭�}{{\mathrm{唯}}_{\mathit{1}}}^{\mathit{*}}{\mathrm{唯}}_{\mathit{2}}\mathrm{dx}=\mathit{-}\frac{i\sout{h}}{\mathit{2}m}{\overline{)\mathit{(}{唯}_{\mathit{2}}\frac{\mathit{鈭�}{唯}_{\mathit{1}}^{\mathit{*}}}{\mathit{鈭�}x}\mathit{-}{唯}_{\mathit{1}}^{\mathit{*}}\frac{\mathit{鈭�}{唯}_{\mathit{2}}}{\mathit{鈭�}x}\mathit{)}}}_{\mathit{鈭�}}^{\mathit{鈭�}} \end{gathered}$$

Where this must equal zero because ${\mathrm{唯}}_1.$and ${\mathrm{唯}}_2$are normalized.

Hence, the solutions for ${\mathrm{唯}}_1.$and ${\mathrm{唯}}_2$is$\frac{\mathrm{d}}{\mathrm{dt}}{鈭�}_{-鈭�}^{鈭�}{\mathrm{唯}}_1^{*}{\mathrm{唯}}_{\mathit{2}}dx=0$