Because we have found no way to formulate quantum mechanics based on a single real wave function, we have a choice to make. In Section 4.3,it is said that our choice of using complex numbers is a conventional one. Show that the free-particle Schrodinger equation (4.8) is equivalent to two real equations involving two real functions, as follows:

_{$\mathbf{-}\frac{{\mathbf{\hslash }}^{\mathbf{2}}{\mathbf{\lambda }}^{\mathbf{2}}{\mathbf{\Psi }}_{\mathbf{1}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}}{\mathbf{\partial }\mathbf{\text{m}}}\mathbf{=}\mathit{\hslash }\frac{\mathbf{\partial }{\mathbf{\Psi }}_{\mathbf{2}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}}{\mathbf{\partial }\mathbf{\text{t}}}$}and

$\mathbf{-}\frac{{\mathbf{\hslash }}^{\mathbf{2}}{\mathbf{\lambda }}^{\mathbf{2}}{\mathbf{\Psi }}_{\mathbf{2}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}}{\mathbf{\partial }\mathbf{\text{m}}}\mathbf{=}\mathit{\hslash }\frac{\mathbf{\partial }{\mathbf{\Psi }}_{\mathbf{1}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}}{\mathbf{\partial }\mathbf{\text{t}}}$

where $\mathit{\Psi }\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}$is by definition ${\mathit{\Psi }}_{\mathbf{1}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}\mathbf{+}\mathbf{\text{i}}{\mathit{\Psi }}_{\mathbf{2}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}$. How is the complex approach chosen in Section4.3more convenient than the alternative posed here?

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$\Psi ={\Psi }_1+i{\Psi }_2$.

_{$\frac{{\hslash }^2}{\text{2m}}\frac{{\partial }^2{\Psi }_1\left(\text{x,t}\right)}{\partial {\text{x}}^2}=\hslash \frac{\partial {\Psi }_2\left(\text{x,t}\right)}{\partial \text{t}}$}

_{$-\frac{{\hslash }^2}{2\text{m}}\frac{{\partial }^2{\Psi }_2\left(\text{x,t}\right)}{\partial {\text{x}}^{\text{2}}}=\hslash \frac{\partial {\Psi }_1\left(\text{x,t}\right)}{\partial \text{t}}$}

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

_{$\frac{{\hslash }^2}{\text{2m}}\left(-i\frac{{\partial }^2{\Psi }_2\left(\text{x,t}\right)}{\partial {\text{x}}^{\text{2}}}-\frac{{\partial }^2{\Psi }_1\left(\text{x,t}\right)}{\partial {\text{x}}^{\text{2}}}\right)=\hslash \left(\text{i}\frac{\partial {\Psi }_1\left(\text{x,t}\right)}{\partial \text{t}}-\frac{\partial {\Psi }_2\left(\text{x,t}\right)}{\partial t}\right)$}

_{$-\frac{{\hslash }^2}{2\text{m}}\left(\frac{{\partial }^2\left({\Psi }_1\left(\text{x,t}\right)+i{\Psi }_2\left(\text{x,t}\right)\right)}{\partial {\text{x}}^2}\right)=\text{i}\hslash \left(\frac{\partial \left({\Psi }_1\left(\text{x,t}\right)+\text{i}{\Psi }_2\left(\text{x,t}\right)\right)}{\partial \text{t}}\right)$}

_{$\Psi ={\Psi }_1+\text{i}{\Psi }_2$}

_{$-\frac{{\hslash }^2}{\text{2m}}\left(\frac{{\partial }^2\Psi }{\partial {\text{x}}^{\text{2}}}\right)=\text{i}\hslash \left(\frac{\partial \Psi }{\partial \text{t}}\right)$}

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