The idea that waves are associated with differential equations was
well-established when Debye suggested to Schrödinger
that he try to find the appropriate equation for deBroglie waves.
Although it is impossible to derive Schrödinger's equation,
it is possible to make plausible assumptions
and then compare the results with experiment. If we assume that a free
particle propagating in the
x-direction can be approximated by a plane wave and adopt deBroglie's
relations for momentum and energy, 
and 
, we
obtain a wavefunction of the form
Substituting eq:PWave in place of the electric field into eq:EME results in the energy-momentum relationship for particles of zero rest mass, i.e. photons.
In order to describe particles of finite rest mass, it is necessary to modify eq:EME to be relativisticly invariant by using Einstein's mass-energy relationship.
These considerations led Schrödinger to the following equation in 1925:
However, this equation was first published by Klein and Gordon and is now known as the Klein-Gordon (KG) equation. It is still used in some text books as an introduction to relativistic wave equations.[14] What we now call the Schrödinger equation
is actually the nonrelativistic limit of eq:KGE.[9]
