The work of Planck, Einstein, and Compton led to the acceptance of the quantum theory of energy that stated light comes in discrete packets of energy called photons and though mostly wave light also has particle properties. Their work inspired Bohr to develop an atomic model explaining the behavior of electrons.
However, Bohr's model was doomed to fail because he modified classical mechanics to make his ideas work. The old Newtonian physics didn't work in the subatomic world, a new physics was needed to solve the problems of the atom and its particles.
In 1924 a French graduate student, Louis de Broglie, was impressed with Einstein's 1905 discovery and the dual nature of light. So impressed, that in his graduate thesis, he suggested that since nature is often symmetric, matter which is mostly particle should also have wave properties. These waves were not electromagnetic but a new kind of wave, matter waves.
Using the equations of Planck, Einstein and Compton, he developed an equation for the wavelength of matter waves, but had no experimental evidence to support his proposal.
But in 1927, the American physicists Clinton J. Davisson and Lester H. Germer verified de Broglie's hypothesis experimentally. They were able to diffract a beam of electrons in a crystal of nickel and only waves can be diffracted. Therefore, if an electron that is particle can be diffracted, it must have wave properties. De Broglie was right, matter has wave properties.
In 1926 Erwin Schrödinger combined the Bohr model with de Broglie's hypothesis.
He proposed the electron was a 3-D waveform circling the nucleus in a whole number of wavelengths allowing the waveform to repeat itself as a stable standing wave representing the energy levels of the Bohr model.
A standing wave is one that does not transfer energy or move but it does undergo resonance. That is to say, it can absorb energy from a nearby source which is oscillating at a proper frequency.
A standing wave must also have a wavelength such that a whole number of wave segments fit within its setting. If the number of wave segments isn't a whole number then the wave will collapse.
Schrödinger suggested that de Broglie was correct about matter waves and the electrons are located in the atomic space according to standing wave frequencies. Therefore, the energy needed to change from one standing wave to another must be quantized in order to maintain a whole number of wavelengths and avoid collapsing.
In support of his hypothesis, Schrödinger developed a mathematical equation to describe the wave-like behavior of the electron. The Schrödinger wave equation not only gave the correct energy levels for the hydrogen atom, but also was somewhat useful in atoms with more than one electron.
This mathematical description of the details of atomic behavior became known as the Wave-mechanical model. I invite you to learn more about chemistry at http://yourCHEMcoach.com or http://mrcausey.info/chemistry