From standing matter waves of the atoms to Schrodinger equation I read in the Resnick - Halliday Physics textbook (2nd edition):

"The idea that stable states in atoms correspond to standing matter waves was adopted by Erwin Schrodinger in 1926 and used by him as the foundation of wave mechanics..."

What was Schrodinger's main reasoning which from stable states of atoms led him to wave mechanics?
 A: 
Adopting a proposal made by Louis de Broglie in 1924 that particles of matter have a dual nature and in some situations act like waves, Schrödinger introduced a theory describing the behaviour of such a system by a wave equation that is now known as the Schrödinger equation. The solutions to Schrödinger’s equation, unlike the solutions to Newton’s equations, are wave functions that can only be related to the probable occurrence of physical events. The definite and readily visualized sequence of events of the planetary orbits of Newton is, in quantum mechanics, replaced by the more abstract notion of probability.

A: It was De Broglies matter wave theory, as did Anna already say. Niels Bohr was furious about the Schrödinger model, because he thought that only his model would correctly explain the atomic resonance spectra (Lyman series aso.) and Schrödinger's model wouldn't, but actually both models are able to calculate it.
If someone wonders why Bohr didn't accept the Schrödinger model is because at first, De Broglies matter waves were a hypothesis at best when Schrödinger created his model. It was only experimentally confirmed afterwards by G.P. Thomson in 1926 and by C. Davisson and L. H. Germer in 1927.
Heisenberg then came with this "quantum egg", the uncertainty principle, in order to prove his tutor Bohr's model right. It's a mathematically correct, but semantically very crappy theory that doesn't make too much sense. But Schrödinger's waves are mostly mathematical too, because he never explained, when calculating the particle in a box, what that "box" really is supposed to be, because there is no box in reality, is there?
After that the thirties started and the scientists like the whole Western society had other, rather political problems to deal with than scientific models and their discussions. That's why these models stalled back then. And string theory is currently stalling too, so "Im Westen nichts Neues".
