Quantum Theory: Louis de Broglie
de Broglie's Discovery of the Wave Properties of Electron Interactions (1927)The next step was taken by de Broglie. He asked himself how the discrete states could be understood by the aid of current concepts, and hit on a parallel with stationary (standing) waves, as for instance in the case of proper frequencies of organ pipes and strings in acoustics. Albert Einstein, 1954)de Broglie's realisation that standing waves exist at discrete frequencies and thus energies is obviously true and important, yet he also continued with the particle concept and thus imagined particles moving in a wavelike manner. These predicted wave properties of matter were shortly thereafter confirmed from experiments (Davisson and Germer, 1927) on the scattering of electrons through crystals (which act as diffraction slits). As Albert Einstein confirms; Experiments on interference made with particle rays have given brilliant proof that the wave character of the phenomena of motion as assumed by the theory does, really, correspond to the facts. (Albert Einstein, 1954)) In 1913, Niels Bohr had developed a simple (though only partly correct) model for the hydrogen atom that assumed; i) That the electron particle moves in circular orbits about the proton particle. ii) Only certain orbits are stable / allowed. iii) Light is emitted and absorbed by the atom when the electron 'jumps' from one allowed orbital state to a another. This early atomic model had some limited success because it was obviously created to explain the discrete energy states of light emitted and absorbed by bound electrons in atoms or molecules, as discovered by Planck in 1900. de Broglie was aware of Bohr's model for the atom and he cleverly found a way of explaining why only certain orbits were 'allowed' for the electron. As Albert Einstein explains; de Broglie conceived an electron revolving about the atomic nucleus as being connected with a hypothetical wave train, and made intelligible to some extent the discrete character of Bohr's 'permitted' paths by the stationary (standing) character of the corresponding waves. (Albert Einstein, 1940)) Fig: 1. The allowed discrete orbits of the electron as imagined by de Broglie. de Broglie assumed that because light had both particle and wave properties, that this may also be true for matter. Thus he was not actually looking for a wave structure of matter. Instead, as matter was already assumed to be a particle, he was looking for wave properties of matter to complement the known particle properties. As a consequence of this particle/wave duality, de Broglie imagined the standing waves to be related to discrete wavelengths and standing waves for certain orbits of the electron particle about the proton. (Rather than considering an actual standing wave structure for the electron itself.) From de Broglie's perspective, and from modern physics at that time, this solution had a certain charm. It maintained the particle  wave duality for BOTH light and matter, and at the same time explained why only certain orbits of the electron (which relate to whole numbers of standing waves) were allowed, which fitted beautifully with Niels Bohr model of the atom. de Broglie further explains his reasoning for the particle/wave duality of matter in his 1929 Nobel Prize acceptance speech; On the one hand the quantum theory of light cannot be considered satisfactory since it defines the energy of a light particle (photon) by the equation E=hf containing the frequency f. Now a purely particle theory contains nothing that enables us to define a frequency; for this reason alone, therefore, we are compelled, in the case of light, to introduce the idea of a particle and that of frequency simultaneously. On the other hand, determination of the stable motion of electrons in the atom introduces integers, and up to this point the only phenomena involving integers in physics were those of interference and of normal modes of vibration. This fact suggested to me the idea that electrons too could not be considered simply as particles, but that frequency (wave properties) must be assigned to them also. (de Broglie, 1929) 
Quantum Theory: Erwin Schrodinger
Schrodinger Wave Equations (1928)Quantum theory is essentially founded on the experimental observations of frequency and wavelength for both light and matter.1. Planck's discovery that energy is related to frequency in the equation E=hf 2. The Equivalence of Energy, Frequency and Mass E=hf=mc2, which deduces the Compton Wavelength Y=h/mc 3. The de Broglie wavelength y=h/mv It was Erwin Schrodinger who discovered that when frequency f and de Broglie wavelength y were substituted into general wave equations it becomes possible to express energy E and momentum mv (from the above equations) as wave functions  thus a confined particle (e.g. an electron in an atom / molecule) with known energy and momentum functions could be described with a certain wave function. From this it was further found that only certain frequency wave functions, like frequencies on musical strings, were allowed to exist. These allowed functions and their frequencies depended on the confining structure (atom or molecule) that the electron was bound to (analogous to how strings are bound to a violin, and only then can they resonate at certain frequencies). Significantly, these allowed frequencies corresponded to the observed discrete frequencies of light emitted and absorbed by electrons bound in atoms/molecules. This further confirmed the standing wave properties of matter, and thus that only certain standing wave frequencies could exist which corresponded to certain energy states. The agreement of observed frequencies and Schrodinger's Wave Equations further established the fundamental importance of Quantum Theory and thus the Wave properties of both light and matter. As Albert Einstein explains; How can one assign a discrete succession of energy values E to a system specified in the sense of classical mechanics (the energy function is a given function of the coordinates x and the corresponding momenta mv)? Planck's constant h relates the frequency f =E/h to the energy values E. It is therefore sufficient to assign to the system a succession of discrete frequency f values. This reminds us of the fact that in acoustics a series of discrete frequency values is coordinated to a linear partial differential equation (for given boundary conditions) namely the sinusoidal periodic solutions. In corresponding manner, Schrodinger set himself the task of coordinating a partial differential equation for a scalar wave function to the given energy function E (x, mv), where the position x and time t are independent variables. (Albert Einstein, 1936) It should also be noted that Schrodinger's wave equations describe scalar waves rather than vector electromagnetic waves. This is a most important difference. Electromagnetic waves are vector waves  at each point in Space the wave equations yield a vector quantity which describes both a direction and an amplitude (size of force) of the wave, and this relates to the original construction of the em field by Faraday which described both a force and a direction of how this force acted on other matter particles. Scalar wave equations yield a single quantity for each point in space which simply describes the wave amplitude (there is no directional component). For example, sound waves are scalar waves where the wave amplitude describes the Motion (or compression) of the wave medium (air). With de Broglie's introduction of the concept of standing waves to explain the discrete energy states of atoms and molecules, and the introduction of scalar waves by Schrodinger, they had intuitively grasped important truths of nature as Albert Einstein confirms; Experiments on interference made with particle rays have given brilliant proof that the wave character of the phenomena of motion as assumed by the theory does, really, correspond to the facts. The de BroglieSchrodinger method, which has in a certain sense the character of a field theory, does indeed deduce the existence of only discrete states, in surprising agreement with empirical facts. It does so on the basis of differential equations applying a kind of resonance argument. (Albert Einstein, 1927) 
Quantum Theory: Heisenberg Born Probability Interpretation
Heisenberg's Uncertainty Principle & Born's 'Probability Waves' Interpretation of Quantum Theory (1928)At the same time that the wave properties of matter were discovered, two further discoveries were made that also profoundly influenced (and confused) the future evolution of modern physics.Firstly, Werner Heisenberg developed the uncertainty principle which tells us that we (the observer) can never exactly know both the position and momentum of a particle. As every observation requires an energy exchange (photon) to create the observed 'data', some energy (wave) state of the observed object has to be altered. Thus the observation has a discrete effect on what we measure. i.e. We change the experiment by observing it! Further, because both the observed position and momentum of the particle can never be exactly known, theorists were left trying to determine the probability of where, for example, the 'particle' would be observed. Born (1928) was the first to discover (by chance and with no theoretical foundation) that the square of the quantum wave equations (which is actually the WaveDensity) could be used to predict the probability of where the particle would be found. Since it was impossible for both the waves and the particles to be real entities, it became customary to regard the waves as unreal probability waves and to maintain the belief in the 'real' particle. This maintained the belief in the particle/wave duality, though in a new form where the 'quantum' scalar standing waves had become 'probability waves' for finding the 'real' particle. Albert Einstein agreed in part with this probability wave interpretation of Quantum Theory, as he believed in continuous force fields (not in waves or particles) thus to him it was sensible that the waves were not real, and were mere descriptions of probabilities. He writes; On the basis of quantum theory there was obtained a surprisingly good representation of an immense variety of facts which otherwise appeared entirely incomprehensible. But on one point, curiously enough, there was failure: it proved impossible to associate with these Schrodinger waves definite motions of the mass points  and that, after all, had been the original purpose of the whole construction. The difficulty appeared insurmountable until it was overcome by Born in a way as simple as it was unexpected. The de BroglieSchrodinger wave fields were not to be interpreted as a mathematical description of how an event actually takes place in time and space, though, of course, they have reference to such an event. Rather they are a mathematical description of what we can actually know about the system. They serve only to make statistical statements and predictions of the results of all measurements which we can carry out upon the system. (Albert Einstein, 1940) It seems to be clear, therefore, that Born's statistical interpretation of quantum theory is the only possible one. The wave function does not in any way describe a state which could be that of a single system; it relates rather to many systems, to an 'ensemble of systems' in the sense of statistical mechanics. (Albert Einstein, 1936) Albert Einstein refers to the now well established fact that matter interacts with other matter throughout the universe. He realised that because matter is somehow spherically spatially extended we must give up the idea of complete localization and knowledge of the 'particle' in a theoretical model. Einstein believed it was this lack of knowledge of the system as a whole that is the ultimate cause of the uncertainty and resultant probability inherent in Quantum Theory. Thus the last and most successful creation of theoretical physics, namely quantum mechanics (QM), differs fundamentally from both Newton's mechanics, and Maxwell's em field. For the quantities which figure in QM's laws make no claim to describe physical reality itself, but only probabilities of the occurrence of a physical reality that we have in view. (Albert Einstein, 1931) I cannot but confess that I attach only a transitory importance to this interpretation. I still believe in the possibility of a model of reality  that is to say, of a theory which represents things themselves and not merely the probability of their occurrence. On the other hand, it seems to me certain that we must give up the idea of complete localization of the particle in a theoretical model. This seems to me the permanent upshot of Heisenberg's principle of uncertainty. (Albert Einstein, 1934)) Albert Einstein believed that Reality could be represented by continuous spherical force fields, that reality was not founded on chance (as Bohr and Heisenberg argued) but on necessary connections between things (thus his comment 'God does not play dice'!). 
Quantum Theory: Famous Quotes
The more success the quantum theory has, the sillier it looks. (Albert Einstein to Heinrich Zangger, May 20, 1912)

based1. http://www.SpaceandMotion.com/PhysicsQuantumTheoryMechanics.htm 