Linear superposition of solutions is a solution of TDSE

[Silversonic]

Homework Statement

Consider two normalised, orthogonal solutions of the TDSE

(Note all my h's here are meant to be h-bar, I'm not sure how to get a bar through them).

[itex]\Psi_1 = \psi_1 (x) e^{-E_1 it/h}[/itex]

[itex]\Psi_2 = \psi_2 (x) e^{-E_2 it/h}[/itex]

Consider the wavefunction

[itex] \Phi = \sqrt{\frac{1}{3}}\Psi_1 + \sqrt{\frac{2}{3}}\Psi_2 [/itex]

Which is also a normalised solution to the TDSE. Any linear superposition of solutions to the TDSE is also a solution.

Homework Equations

The TDSE is;

[itex]\widehat{H}\Phi = ih\frac{\delta}{\delta t}\Phi[/itex]

The Attempt at a Solution

This isn't a question, I've just gone to pursue the statement that the linear superposition is also a solution. I can't see to show it though.

Using [itex] \Phi = \sqrt{\frac{1}{3}}\Psi_1 + \sqrt{\frac{2}{3}}\Psi_2 [/itex]

[itex]\widehat{H}\Phi = ih\frac{\delta}{\delta t}\Phi = \sqrt{\frac{1}{3}}E_1\psi_1 (x) e^{-E_1 it/h} + \sqrt{\frac{2}{3}}E_2\psi_2 (x) e^{-E_2 it/h}[/itex]

Which I cannot manage to get into the form

[itex] (\sqrt{\frac{1}{3}}E_1 + \sqrt{\frac{2}{3}}E_2)\Phi [/itex]

Which is the form is should be in if it was a solution surely?

 

[ehild]

Any linear combination of two solutions is also a solution of the TDSE. But the linear combination of two eigenfunctions is not an eigenfunction if functions belonging to different energies are involved.

Show that Φ is a solution by substituting into the TDSE.

Show that it is normalized.

ehild