Quantum Mechanics as Classical Continuum Physics and Not Particle Mechanics

Planck (with eyes shut) presents Einstein with the Max Planck medal of the German Physical Society, 28 June 1929, in Berlin, as the highest award of the Deutsche Physikalische Gesellschaft, for Einstein's idea of light as particles, which Planck did not believe in (and did not want to see).

Modern physics in the form of quantum mechanics was born in 1900 when Planck in a desperate act introduced the idea of smallest packet of energy or quanta to explain black-body radiation followed up in 1905 by Einstein's equally desperate attempt to explain photo-electricity by viewing light as a stream of light particles of energy quanta $h\nu$ where $\nu$ is frequency and $h$ Planck's constant.

Yes, Einstein was desperate, because he was stuck as patent clerk in Bern and his academic career was going nowhere. Yes, Planck was also desperate because his role at the University of Berlin as the successor of the great Kirchhoff, was to explain blackbody radiation as the most urgent unsolved problem of physics and thereby demonstrate the scientific leadership of an emerging German Empire.

The "quantisation" into discrete smallest packets of energy and light  was against the wisdom of the continuum physics of the 19th century crowned by Maxwell's wave equations describing all of electro-magnetics as a system of partial differential equations over 3d-space as a continuum over real numbers as the ultimate triumph of the infinitesimal Calculus of Leibniz and Newton.

The "quantisation" of energy and light thus meant a partial retreat to the view of the early Greek atomists with the world ultimately built from indivisible particles or quanta and not waves, also named particle physics.

But the wave nature was kept in Schrödinger's linear multi-d equation as the basis of quantum mechanics, but then not in physical form as in Maxwell's equations, but as probability waves supposedly describing probabilities of particle configurations. The mixture was named wave-particle duality, which has been the subject of endless discussion after its introduction by Bohr.

Schrödinger never accepted a particle description and stuck to his original idea that waves are enough to explain atom physics. The trouble with this view was the multi-d aspect of Schrödinger's equation which could not be given a meaning/interpretation in terms of physical waves, like Maxwell's equations. This made Schrödinger's waves-are-enough idea impossible to defend and Schrödinger's equation was hijacked Bohr/Born/Heisenberg and twisted into a physical particle - probabilistic wave Copenhagen Interpretation as the textbook truth.

But blackbody radiation and the photoelectric effect can be explained by wave mechanics  without any form of particles in the form of Computational Blackbody Radiation with the new element being finite precision computation. 

The idea of a particle  is contradictory, as something with physical presence without physical dimension. Atom physics can make sense as wave mechanics but not as particle mechanics. It is important to remember that this was the view of Schrödinger when he formulated his wave equation in 1925 for the Hydrogen atom. What is needed is an extension of Schrödinger's equation to atoms with several electrons which has a physical meaning, maybe as Real Quantum Mechanics, and this is not the standard linear multi-d Schrödinger equation with solutions interpreted as probability distributions of particle configurations in the spirit of Born-Bohr-Heisenberg but not Schrödinger.

Recall that particle motion is also a contradictory concept, as shown in Zeno's paradox: At each instant of time the particle (Zeno's arrow) is still at a point in space, and thus cannot move to another point. On the other hand, wave motion as the translatory motion of a water wave across a water surface of water, is possible to explain as the result of (circular) transversal water oscillation without translation. Electro-magnetic waves are  propagating by transversal oscillation of electric-magnetic fields.

And do not believe that Zeno's paradox was ever solved. It expresses the truly contradictory nature of the concept of particle, which cannot be resolved. Ponder the following "explanation" on Stanford Encyclopedia of Philosophy:

• Think about it this way: time, as we said, is composed only of instants. No distance is traveled during any instant. So when does the arrow actually move? How does it get from one place to another at a later moment? 
• There's only one answer: the arrow gets from point X at time 1 to point Y at time 2 simply in virtue of being at successive intermediate points at successive intermediate times—the arrow never changes its position during an instant but only over intervals composed of instants, by the occupation of different positions at different times. 
• In Bergson's memorable words—which he thought expressed an absurdity—‘movement is composed of immobilities’ (1911, 308): getting from X to Y is a matter of occupying exactly one place in between at each instant (in the right order of course). 

As you understand, this is just nonsense:

Particles don't exist, and if they anyway are claimed to exist, they cannot move.

Waves do exist and can move.  It is not so difficult to understand!