[Physics] Non-separable solution for the Schrödinger equation

quantum mechanicsschroedinger equationwavefunction

Schrödinger solutions are usually if not always of the type: $\psi=\operatorname{T}(t)*\operatorname{X}(x)$ (we use the separation of variables method to arrive at the time independent Schroedinger equation).

I was trying to find a non-separable solution. To this purpose I tried the following method: compose the product T(t)*X(x) into another function. For example: $\sinh(\operatorname{T}(t) \cdot \operatorname{X}(x))$ or $\ln(\operatorname{T}(t) \cdot \operatorname{X}(x))$ or $(\operatorname{T}(t) \cdot \operatorname{X}(x))^2$ and so on.

Then I try first time derivative of $\psi = \mathrm{second}$ position derivative of $\psi$ (trying to find a solution for the null potential. In addition I know I need a constant but it's for simplification).

I arrive at a differential equation. I try simple solutions for one of the functions like $\operatorname{T}(t)=t$. I get to a very difficult differential equation that can't be solved even with a
look at:
http://www.amazon.com/Handbook-Solutions-Ordinary-Differential-Equations/dp/1584882972

If there any obvious non separable solution that I am missing?

Best Answer

Well, solutions of the Schroedinger equation are usually (but not always) written in the form $\psi = \sum_n T_n(t) \ast X_n(x)$ -- i.e. as a sum of separable solutions. In fact, it is always possible to write the solution in this form. It follows from the fact that the Hamiltonian is Hermitian and all Hermitian operators are diagonalizable. (I'm ignoring operators with continuous spectra, but this is not a serious complication.)

Solutions which are not of the form $T(t)\ast X(x)$ are still sometimes useful though -- coherent states are one example.

I can give more details on any of the above if you ask.