Question

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?

Short answer

We should be able to recover the original Schrodinger equation using the two supplied equations, using the information that_{$\Psi ={\Psi }_1+i{\Psi }_2$}.

Step-by-step solution

Combining two partial differential equations.

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.

Conclusion

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