No, We Don't Understand Quantum Mechanics, But There Is Hope.

                                                            Yes, QM is a strange world.

The Preface of book Do We Really Understand Quantum Mechanics by Franck Laloe supplemented by an article with the same title, tells the truth about quantum mechanics:

• In many ways, quantum mechanics QM is a surprising theory... because it creates a big contrast between its triumphs and difficulties. 
• On the one hand, among all theories, quantum mechanics is probably one of the most successful achievements of science.  The applications of quantum mechanics are everywhere in our twentyfirst century environment, with all sorts of devices that would have been unthinkable 50 years ago. 
• On the other hand, conceptually this theory remains relatively fragile because of its delicate interpretation – fortunately, this fragility has little consequence for  its efficiency. 
• The reason why difficulties persist is certainly not that physicists have tried to ignore them or put them under the rug!
• Actually, a large number of interpretations have been proposed over the decades, involving various methods and mathematical techniques. 
• We have a rare situation in the history of sciences: consensus exists concerning a systematic approach to physical phenomena, involving calculation methods having an extraordinary predictive power; nevertheless, almost a century after the introduction of these methods, the same consensus is far from being reached concerning the interpretation of the theory and its
  foundations. 
• This is reminiscent of the colossus with feet of clay.
• The difficulties of quantum mechanics originate from the object it uses to describe physical systems, the state vector (wave function) $\Psi$.
• Without any doubt, the state vector is a curious object to describe reality!

The message is that QM a formidable achievement of the human intellect which is incredibly useful in practice, but like a colossus with feet of clay has a main character flaw, namely that it is a curious way to describe reality and as such not understood by physicists. 

There are two ways the handle if a physical theory is not understood because it is so curious, either the theory is dismisssed as being seriously flawed or the curiosity is chosen as a sign that the theory is correct and beyond questioning by human minds.

The reason QM is so mysterious is that the wave function $\Psi =\Psi (x_1,x_2,…,x_N)$ for an atom or molecule with $N$ electrons depends on $N$ independent three-dimensional space variables $x_1$, $x_2$,…, $x_N$, together with time, thus is a function in $3N$ space dimensions plus time and as such has no direct real physical meaning since real physics takes place in $3$ space dimensions. 

The wave function $\Psi$ is introduced as a the solution to a linear multi-dimensional linear wave equation named Schrödinger's equation of the form

• $i\frac{\partial\Psi}{\partial t}+H\Psi = 0$,

where $H$ is a Hamiltonian operator acting on wave functions. The mysticism of QM thus originates from Schrödinger's equation and is manifested by the fact that there is no real derivation of Schrödinger's equation from basic physical laws. Instead, Schrödinger's equation is motivated as a purely formal manipulation of classical Hamiltonian mechanics without physical meaning. 

The main trouble with QM based on a linear multi-d Schrödinger equation is thus the physical interpretation of the multi-d wave function and the accepted answer to this enigma is to view 

• $\vert\Psi (x_1,…,x_N)\vert^2$ 

as a probability distribution of a particle configuration described by the coordinates $(x_1,…,x_N)$ representing human knowledge about a physics and not physics itself. Epistemology of what we can know is thus allowed to replace ontology of what is.

The linear multi-d Schrödinger equation thus lacks connection to physical reality. Moreover, because of its many dimensions the equation cannot be solved (analytically or computationally), and the beautiful net result is that QM is based on an equation without physical meaning which cannot be solved. No wonder that physicists still after 100 years of hard struggle do not really understand QM. 

But since Schrödinger's linear multi-d equation lacks physical meaning (and neither can be solved) there is no compelling reason to view it as the foundation of atomistic physics. 

It appears to be more constructive to consider instead systems of non-linear Schrödinger equations in $N$ three-dimensional wave functions $\psi_1(x),…,\psi_N(x)$ with $x$ a 3d space coordinate,  in the spirit of of Hartree models, as physically meaningful computable models of potentially great practical usefulness. 

Sums of such wave functions then play a basic role and have physical meaning, to be compared the standard setting with $\Psi (x_1,…,x_N)$ in the form of Slater determinants as sums of muli-d products $\psi (x_1)\psi (x_2)…\psi (x_N)$ of complicated unphysical nature.