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 
 A: 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.
A: Regarding your question about the generality of separation of variables, there is an extremely beautiful Lie-theoretic approach to symmetry, separation of variables and special functions,
e.g. see Willard Miller's book [1]. I quote from his introduction:

This book is concerned with the
  relationship between symmetries of a 
  linear second-order partial
  differential equation of mathematical
  physics,  the coordinate systems in
  which the equation admits solutions
  via separation  of variables, and the
  properties of the special functions
  that arise in  this manner. It is an
  introduction intended for anyone with
  experience in  partial differential
  equations, special functions, or Lie
  group theory, such  as group
  theorists, applied mathematicians,
  theoretical physicists and  chemists,
  and electrical engineers. We will
  exhibit some modern group-theoretic 
  twists in the ancient method of
  separation of variables that can be 
  used to provide a foundation for much
  of special function theory. In 
  particular, we will show explicitly
  that all special functions that arise 
  via separation of variables in the
  equations of mathematical physics can 
  be studied using group theory. These
  include the functions of Lame, Ince, 
  Mathieu, and others, as well as those
  of hypergeometric type. 
This is a very critical time in the
  history of group-theoretic methods in 
  special function theory. The basic
  relations between Lie groups, special 
  functions, and the method of
  separation of variables have recently
  been  clarified. One can now construct
  a group-theoretic machine that, when 
  applied to a given differential
  equation of mathematical physics,
  describes  in a rational manner the
  possible coordinate systems in which
  the equation  admits solutions via
  separation of variables and the
  various expansion  theorems relating
  the separable (special function)
  solutions in distinct  coordinate
  systems. Indeed for the most important
  linear equations, the  separated
  solutions are characterized as common
  eigenfunctions of sets of 
  second-order commuting elements in the
  universal enveloping algebra of  the
  Lie symmetry algebra corresponding to
  the equation. The problem of 
  expanding one set of separable
  solutions in terms of another reduces
  to a  problem in the representation
  theory of the Lie symmetry algebra.

For an example of effective Lie-theoretic algorithms for first-order ODEs see Bruce Char's paper[2], from which the following useful tables are extracted.
 

1 Willard Miller. Symmetry and Separation of Variables.
Addison-Wesley, Reading, Massachusetts, 1977 (out of print)
http://www.ima.umn.edu/~miller/separationofvariables.html
http://gigapedia.com/items:links?id=64401
2 Bruce Char. Using Lie transformation groups to find closed form solutions to first order ordinary differential equations.
SYMSAC '81. Proceedings of the fourth ACM symposium on Symbolic and algebraic computation.
http://portal.acm.org/citation.cfm?id=806370 
