Two of the aforementioned failures of classical physics lead directly to the earthshaking matter-wave hypothesis of deBroglie: Einstein had demonstrated the wave-particle duality of light through his theory of the photon and Bohr, through his solar-system model of the hydrogen atom, hinted at the quantization of electron energies (we will cover his model more thoroughly when we get to atoms but the basics are well known to anyone who has taken general chemistry). Using a few equations from special relativity, deBroglie not only assumed that all matter had wavelike (as well as particle) properties, but in 1923 he derived a very simple relationship between wavelength and momentum: λ=h/p. In doing so, he essentially bridged what is now called the "old quantum theory", which lasted for about twenty years, to the much more successful (and abstract and perhaps unsettling) quantum mechanics of Schrödinger and Heisenberg.
The mathematical description of traveling and standing waves, utilizing terms like amplitude, frequency, wavelength, wavevector and period, had already been well-established by the mid-19th century. The Schrödinger Equation, though invoking these same mathematical constructs, can not be derived from any theory: it is simply asserted as a complete description of the physics of small particles. When the potential experienced by a particle of mass m is time-independent, we get the following equation (which should be committed to memory for any student studying quantum mechanics):
What changes from problem to problem is the nature of the potential V and the coordinate system, which will change the form of the laplacian operator and what is obtained when the Schrodinger Equation is solved are mathematical descriptions of the wavefunction ψ and the energy E. We will soon see that ψ contains all the possible information about the quantum system but what is not so trivial is getting that information out. For that we will need the ideas of operator alebra and eigenvalue equations.
Next up: Descriptions of the momentum and position operators, followed by the very important hamiltonian operator, and our first solution to the Schrödinger Equation, the so-called particle in a box problem.