Separation of Schrodinger Equation

[NBaca]

Homework Statement

Show that the time-dependent Schrödinger equation is separable when V depends on time only and is uniform in space (i.e., V = V (t)).

Homework Equations

The Attempt at a Solution

In the attached document

 

[vela]

Homework Statement

Show that the time-dependent Schrödinger equation is separable when V depends on time only and is uniform in space (i.e., V = V (t)).

Your work is fine up to here:

$$-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}=E\psi$$

$$i\hbar\frac{1}{\Theta}\frac{\partial \Theta}{\partial t}-V(t)=E$$

You made a simple algebra mistake in the next step of the theta equation. Your attempts to solve the differential equations are seriously misguided. These are pretty basic differential equations. You should review how to solve them.