We have elsewhere pointed out the connection between Fermat's work (in the 17th century) on the propagation of light waves, where he showed that the path chosen was one of "least time" and the work of Hamilton (in the 19th century) on the propagation of particles, where the path chosen was one of "least action". Both situations can be recast to be a path of "least phase length", but only if we assume particles have wave properties. Hamilton did not have any experimental evidence to prod him in that direction, but Schrödinger did, and he started from Hamilton's equations of motion and forced them to accept a wave function. The Hamiltonian of a system is the sum of the kinetic and potential energy and is given the symbol H. The famous Schrödinger Equation is:

Here Y is the symbol for the wavefunction and E is its energy. (Do you know what is meant by the variable "r" here?. And what is that funny looking symbol ? An upside-down delta? And what kind of an equation is this?) This equation holds for a particle in a bound, time-independent state. The time-dependent form of the equation is:

This deals with a particle whose energy is changing with time. Both equations are derived without recourse to relativity - that is they are not Lorentz invariant. This can be seen since the spatial coordinates appear as second derivatives and time appears as a first derivative - the two are not being treated equally. But it works very well for most systems. Schrödinger first tried to write down the Lorentz-invariant equations but was unable to, so he settled for this form. Not until two years later was Paul Dirac able to provide the relativistic equation.

If you look at the Hamiltonian, you can start to appreciate what has been done, in moving from classical Hamiltonian theory.