Quantum Mechanics Without Quantisation

Schrödinger's Equation SE for the Hydrogen atom with one electron has the form of a classical continuum mechanical wave equation in a complex-valued wave function $\psi (x,t)$ depending on a 3d space coordinate $x$ and a time coordinate $t$ with $\vert\psi (x,t)\vert^2$ assigned the clear physical meaning of electron charge density at $(x,t)$ with total charge of one unit. The model captures the observed spectrum of Hydrogen as a discrete set of eigenvalues of normalised eigenfunctions in fully classical continuum mechanical form. 

Yet this model has been taken as starting point for a fundamental reformation of classical physics into a fundamentally new form of physics named quantum mechanics resulting from a process of quantisation. In the case of the Hydrogen atom this radical step reduces to a reinterpretation of $\vert\psi (x,t)\vert^2$ as a probability density thus replacing charge density (with physical meaning) with probability (without physical meaning). In this case the reformation makes no sense: The Emperor's New Clothes. Smallest quantum of energy has no physical meaning. 

The reason for the reformation appeared along with the generalisation of SE to atoms with more than one electron, which was the problem facing Schrödinger in 1926 after formulating SE for the Hydrogen atom with one electron, which propelled him to fame. But it was not evident how to proceed and so Schrödinger gave in to a purely formal generalisation introducing a new set of 3d spatial variables for each new electron forming a multi-d SE with only probabilistic interpretation possible and as such aggressively promoted by Bohr-Born-Heisenberg overpowering Schrödinger's request for real physics as ontology instead of unphysical probability as epistemology.

So was the modern physics of quantum mechanics born from a formal process of quantisation, which boiled down to replacing classical deterministic continuum physics by probabilistic physics without determinism and physical meaning. Schrödinger deeply regretted ever to be involved in this project forming 20th century physics. 

Could history have taken a different route by a different generalisation staying within classical continuum physics if Schrödinger had just resisted the onslaught from Bohr-Born-Heisenberg at bit longer? Yes, this would have been possible if only Schrödinger had tried the idea of Real  Quantum Mechanics RealQM of forming a SE in terms of non-overlapping charge densities with direct physical meaning and without any need of reformation by quantisation into probabilities. 

RealQM offers a model of atomic physics in the form of classical continuum physics without any need of quantisation and probabilities. RealQM combines seamlessly with classical electro-magnetics and Newtonian mechanics and so opens to the formation of a Unified Field Theory UFT, which both Schrödinger and Einstein struggled to find throughout the later halfs of their scientific lives, but couldn't do.....Schrödinger died in Vienna in 1961 73 years old... 

Schrödinger in his Nobel Lecture 1933 showing his resistance to Bohr-Born-Heisenberg:

• We cannot, however, manage to make do with such old, familiar, and seemingly indispensible terms as "real" or "only possible"; we are never in a position to say what really is or what really happens, but we can only say what will be observed in any concrete individual case. 
• Will we have to be permanently satisfied with this. . . ? On principle, yes. On principle, there is nothing new in the postulate that in the end exact science should aim at nothing more than the description of what can really be observed. 
• The question is only whether from now on we shall have to refrain from tying description to a clear hypothesis about the real nature of the world. 
• There are many who wish to pronounce such abdication even today. But I believe that this means making things a little too easy for oneself.

ChatGPT about Schrödinger's struggle find a UFT:

• After inventing wave mechanics, Schrödinger spent decades searching for a unified continuum field theory of matter and forces, resisting the idea that nature is fundamentally quantised — but his attempts never succeeded against the empirical dominance of quantum field theory.

• Goal: Matter = continuous wave fields, not particles.

• Method:

  • Original 1926 wave mechanics: electrons as standing waves.

  • Later: attempts to merge wave mechanics with Einstein’s relativity → affine field theory, complex scalar fields.

• Belief: Quantisation is not fundamental, but an artifact of wave modes and stability conditions.

• Outcome: His “unified field theory” never matched experiments; the community rejected it once QED and QFT succeeded.

• Spirit: Continuity is real, discreteness is emergent.