Physics and Mathematics of Schrödinger's Equation

The Schrödinger equation describes the ground state of the Hydrogen atom by the wave function $\Psi (x)$ with $x$ a 3d spatial variable, which minimises the total energy

• $E =E_{kin} + E_{pot}$

as the sum of 

• $E_{kin} =\frac{1}{2}\int\vert\nabla\Psi (x)\vert^2dx$     (electronic kinetic energy)
• $E_{pot} =  -\int\frac{\Psi^2(x)}{\vert x\vert}dx$              (electronic Coulomb potential energy)

under the side condition

•  $\int\Psi^2(x)dx =1.$

The model contains the following three components as functions of $x$: 

1. Distributed charge density: $\Psi^2(x)$ with unit total charge.
2. Distributed kinetic energy: $\vert\nabla\Psi (x)\vert^2$.
3. Distributed potential energy: $-\frac{\Psi^2(x)}{\vert x\vert}$.  

The solution can be computed analytically to be $\Psi (x)=\frac{1}{\sqrt{\pi}}\exp(-\vert x\vert )$. The total energy represents the ground state energy of a Hydrogen atom with kernel at $x=0$. The Coulomb potential is classical physics, while the kinetic energy is a new form of energy measured by the gradient $\nabla\Psi (x)$ as an analog to classical elastic energy. The model has a clear physical meaning and the ground state is characterised by a charge density which concentrates around the kernel paying a kinetic energy cost. 

The Schrödinger equation for the Hydrogen atom in charge density is an example of an Eulerian continuum model of the same form as the Navier-Stokes equations for fluid flow in terms of velocity and pressure as distributed functions of a 3d space coordinate, where individual particle trajectories are not followed. A major advantage of a continuum model is that it allows very efficient computation under discretisation of different spatial resolution.

Schrödinger's equation does not involve point positions of electrons, just distributed charge density, and thus has nothing to say about point positions of electron. At least this is what a mathematician would say understanding that a mathematical model does not contain more than what is put in. Even an emergent phenomenon is a consequence of input. It is not meaningful to ask about point positions of the planets in the Solar system at some given time from a model that only contains time-less orbits of the planets. 

A mathematician would add that neither can exact point position of the electron of a Hydrogen atom be determined experimentally. Wittgenstein would agree that asking about electron position does not make sense in the Schrödinger charge density model and so should not no be spoken of.  

But physicists would not hesitate to say that it is meaningful to ask about the position of the Hydrogen electron, even if it is not contained in the model and cannot be experimentally determined. A physicist would insists that Schrödinger's equation is to be viewed to be formulated in terms of a probability density of electron point position, and not charge density. 

But a probability density is not by itself any physical quantity, and is instead by physicists described as a catalog of possible electron point positions. But physics does not keep such a catalog. Changing from charge density to probability density thus turns Schrödinger's equation from being a model of physics into a model of non-physics. 

So why did physicists take this step in 1926 when seeking to give a meaning to Schrödinger's equation for the Hydrogen atom, which had shown to accurately capture the spectrum of the Hydrogen atom from electronic energies of excited states of the ground state. Schrödinger certainly viewed his equation for Hydrogen in terms of charge density and not probability density.

The switch to probability density came with an ad hoc generalisation of Schrödinger's equation for atoms with more than one electron in terms of a multi-dimensional wave function depending on $3N$ spatial variables for an atom with $N>1$ electrons. With the help of Max Born this model was given an interpretation in terms of probability density to form Standard Quantum Mechanics StdQM as the foundation of modern physics. Here Born's rule states that experimental observation of a prepared quantum state as a given linear combination of eigen-states, choses one of the eigen-states with probability measured by its coefficient in the linear combination. 

This is called collapse of the wave function and has remained a true mystery from physical point of view. The eigen-states are deterministically determined by Schrödinger's equation and so the spectrum. Statistics thus enters only in experimental observation of prepared states while the spectrum is always the same. 

This connects to the view of Bohr that the objective of StdQM is to predict outcomes of prepared experiments, not to model reality. This is mind-boggling and does not seem to make any sense. 

 

Main efforts have been made over the 100 years since 1926 to give StdQM back some physical meaning but no consensus about interpretation in terms of charge density has been reached. 

RealQM presents a new different generalisation to $N>1$ of the Schrödinger equation for $N=1$ in terms of a system of electronic non-overlapping charge densities, which keeps the physicality of the Hydrogen atom. RealQM is a continuum model in 3d with computational cost scaling linearly with $N$, compared to StdQM with exponential scaling. 

The linearity of the Schrödinger equation of StdQM invited to a mathematical analysis using the machinery of the new field of functional analysis developed by Hilbert in terms of Hilbert spaces, which von Neumann exploited in a form of axiomatic formal mathematics (in uncomputable form) with axioms without clear physical meaning.   

Mathematics can thus serve to keep physicality of quantum mechanics as in RealQM, but also by von Neumann abstraction leave physicality as in StdQM. Physics education has been locked on StdQM with all its complications from non-physicality as expressed by Nobel Laureate Murray Gell-Mann 50 years ago:

• Niels Bohr brainwashed a whole generation of theorists into thinking that the job of interpreting quantum theory was done 50 years ago.

The brainwash has continued since text-book physics still today is StdQM. Attempts have been made to give StdQM physical meaning like Many-Worlds and Bohmian Pilot Wave but are not viewed to be successful. RealQM opens a new way of thinking, which apparently has been brainwashed away for 100 years... 

Bohr quotes:

• Physics is not about how the world is, it is about what we can say about the world.
• Those who are not shocked when they first come across quantum theory cannot possibly have understood it.

Yes, it is shocking to learn that StdQM it is not about how the world is, that StdQM is non-physical. Schrödinger would have welcomed RealQM as a theory about physics, while such a thing would have shocked Bohr again...

RealQM is based on the same physics as the Hydrogen atom: Coulomb potential energy and kinetic energy. RealQM appears to be about how the microscopic world is...just like Newton's mechanics based on Newton's laws of motion appears to describe how the macroscopic world is...and RealQM connects seamlessly to Newton's mechanics in a unified continuum model.