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?