Quantum Theory / Mechanics: de Broglie's Discovery of the Wave Properties of Electrons (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)

It is with some frustration that I now read these quotes, as it is obvious in hindsight as to their errors, and how simply they can now be solved! de Broglie's realisation that standing waves exist at discrete frequencies and thus energies is obviously true and important, yet he continued with the error of the particle concept and thus imagined particles moving in a wavelike manner! Nonetheless, as he was close to the truth he had considerable success with his theory, and 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)

So by 1927 the wave properties of matter had been predicted theoretically by de Broglie, and then confirmed by experiment. But unfortunately these scientists continued to believe in the existence of discrete particles, and thus they misinterpreted this most important discovery of the standing wave properties of matter.

de Broglie's Interpretation of the Standing Waves as the Wave-Like Motion of a Particle in Orbit (1927)

In 1913, Niels Bohr had developed a simple (though only partly correct) model for the hydrogen atom that assumed; (Our further comments in brackets)
i) That the electron particle moves in circular orbits about the proton particle. (This is nearly correct, they are not 'orbits' but complex Standing Wave patterns)
ii) Only certain orbits are stable. (This is nearly correct, only certain Standing Wave patterns are resonantly stable)
iii) Light is emitted and absorbed by the atom when the electron 'jumps' from one allowed orbital state to a another. (This is nearly correct, the electrons move from one stable Standing Wave pattern to another. This is known as 'Resonant Coupling' and is explained in Section 1.4.)

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)

 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 the 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 the actual standing wave structure of 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)

The solution to their problems was first found by Wolff (1986). He discovered two things (both of which deserve a Nobel prize in their own right);
Firstly, from reading Feynman's PhD thesis (see reference, Feynman and Wheeler, 1945) he was aware of Feynman's conception of charged particles which 'somehow' generated Spherical Electromagnetic In and Out Waves (Feynman called them advanced and retarded waves), but Wolff realised that there are no solutions for spherical vector electromagnetic waves (which are mathematical waves which require both a quantity of force and a direction of force, i.e. vector). Wolff had the foresight to try using real waves, which are Scalar (defined by their Wave-Amplitude only).
And this then led to a series of remarkable discoveries.
He realized that spherical In and Out-Waves removed the need for a separate particle, as the Wave-Center of the Spherical Waves created the particle effect.
He then discovered that when one spherical standing wave was moving relative to another the Doppler shifts gave rise to BOTH the de Broglie Wavelength AND the Mass increase of Albert Einstein's Relativity. (i.e. Wolff demonstrated that when two charged particles (Wave-Centers of two SSWs) are moving relative to one another they gives rise to beats of interference (caused by the Doppler shifting of the In and Out Waves due to relative Motion) which were identified in experiments as the de Broglie wavelength y=h/mv, and also gave rise to the frequency increases and thus energy/mass increases (as E=hf =mc²) of Special Relativity.
Thus in the one equation he had deduced, with mathematical certainty, the two observed phenomena due to relative motion, which respectively found central parts of both Quantum Theory and Albert Einstein's Special Relativity. (Thus for the first time uniting these two theories from one common theoretical foundation!)
This then led to his further work on resonant coupling which finally solved the puzzle of the 'photon' and explained why light energy is only ever emitted and absorbed in discrete amounts. (See Section 1.4)

Unfortunately for modern physics, and ultimately for human knowledge, this obvious solution was never considered by de Broglie, Albert Einstein, Bohr, Schrodinger, Heisenberg, Dirac, Born, Feynman, etc. etc. Thus the now obvious solution of realising that matter was a Spherical Standing Wave that causes the point particle effect at the Wave-Center remained unknown and ignored, and instead, the confusing and paradoxical concept of the particle / wave duality was retained.