what is separation of variables I was pushing my way through a physics book when the author separated the variables of the Schrödinger equation and I lost the plot:
$$\Psi (x, t) = \psi (x) T(t)$$
can someone please explain how this technique works and is used? It can be in general maths or in the context of this problem. Thanks 

Some functions (not all) $\psi (x,t)$ can be written as a product of a function of $x$ and another function of $t$.  For example, $\psi (x,t)=xt$ can be, while $\psi_2 (x,t)=x^2+t^2$ cannot.  The author is guessing that this will yield a solution to the problem and will go on to show that it does.  After some manipulation the equation comes to something like $f(x)=g(t)$ where the left does not depend on $t$ and the right does not depend on $x$.  Then you argue that since the left does not depend on $t$, the right really doesn't either, and both sides must equal some constant.  So now you are solving $f(x)=g(t)=c$.  As equations in single variables it is usually easier.  Proving that this yields a solution is easy.  Proving that all solutions come as a linear combination of solutions of this form is harder.
