91影视

So, as suggested in the section, combining the two partial differential equations into a single equation should greatly simplify our analysis. We should be able to reconstruct the original Schrodinger equation using the two supplied equations and the information that$唯={唯}_1+i{唯}_2$.

_{$\frac{{\mathrm{鈩�}}^2}{\text{2m}}\frac{{鈭�}^2{唯}_1\left(\text{x,t}\right)}{鈭�{\text{x}}^2}=\mathrm{鈩�}\frac{鈭�{唯}_2\left(\text{x,t}\right)}{鈭�\text{t}}$}

_{$-\frac{{\mathrm{鈩�}}^2}{2\text{m}}\frac{{鈭�}^2{唯}_2\left(\text{x,t}\right)}{鈭�{\text{x}}^{\text{2}}}=\mathrm{鈩�}\frac{鈭�{唯}_1\left(\text{x,t}\right)}{鈭�\text{t}}$}

Subtracting equation(1) from the equation (2), after multiplying it by i with equation(2).

_{$\frac{{\mathrm{鈩�}}^2}{\text{2m}}\left(-i\frac{{鈭�}^2{唯}_2\left(\text{x,t}\right)}{鈭�{\text{x}}^{\text{2}}}-\frac{{鈭�}^2{唯}_1\left(\text{x,t}\right)}{鈭�{\text{x}}^{\text{2}}}\right)=\mathrm{鈩�}\left(\text{i}\frac{鈭�{唯}_1\left(\text{x,t}\right)}{鈭�\text{t}}-\frac{鈭�{唯}_2\left(\text{x,t}\right)}{鈭�t}\right)$}

_{$-\frac{{\mathrm{鈩�}}^2}{2\text{m}}\left(\frac{{鈭�}^2\left({唯}_1\left(\text{x,t}\right)+i{唯}_2\left(\text{x,t}\right)\right)}{鈭�{\text{x}}^2}\right)=\text{i}\mathrm{鈩�}\left(\frac{鈭�\left({唯}_1\left(\text{x,t}\right)+\text{i}{唯}_2\left(\text{x,t}\right)\right)}{鈭�\text{t}}\right)$}

_{$唯={唯}_1+\text{i}{唯}_2$}

_{$-\frac{{\mathrm{鈩�}}^2}{\text{2m}}\left(\frac{{鈭�}^2唯}{鈭�{\text{x}}^{\text{2}}}\right)=\text{i}\mathrm{鈩�}\left(\frac{鈭�唯}{鈭�\text{t}}\right)$}

This is the same equation as in the section 4.3, we only have to solve one differential equation (for a complex function) rather than two differential equations (for real functions), which is a significant simplification given the cost of complexification.

Using $唯={唯}_1+i{唯}_2$we should be able to recover the original Schrodinger equation.