# Separation of variables to solve Schrodinger equations

1. Apr 12, 2013

### ace1719

I've found many articles online that explain how to solve the Schrodinger equation for a potential dependent on x, but not for one dependent on t. A couple articles said that you could not use separation of variables to solve the Schrodinger equation with a time dependent potential, but they did not explain why. Why can you not use separation of variables to solve the Schrodinger equation with a time dependent potential, specifically; V(t)=A*cos(ωt), where A is a constant potential and ω is the angular frequency. Thanks!

2. Apr 12, 2013

Technically, a time-dependent field is not a 'potential' field. The S equation is soluble by separation of variables because the equation can be arranged so that the time-dependence is entirely one one side of the = sign. The other side of the = sign has no time-dependence. Mathematically, this can happen if and only if both the expressions are equal to a constant. In physical terms this constant works out to be the total energy. The wave represents a solution to an oscillator system.
If a time-dependent field is present, the solution is time-dependent and represents a *driven* oscillator, with energy being exchanged between the system and the environment. The source/sink of this variable energy is the time-dependent field.
Specifically, if the time-dependent potential is applied as an operator to a spatially-dependent wave, the solution to the S Equation would require Green's Functions, and the orthogonal coordinates needed for separating the variables would be mixed coordinates in space and time. They would represent nodes in the space/time-dependent wave solution.

3. Apr 14, 2013

### ace1719

Thanks a lot for the explanation, but just to clarify, is there nothing mathematically wrong with doing separation of variables with a time dependent potential?

4. Apr 15, 2013