Triumph of Mathematics: Newton and Schrödinger

In the previous post we discussed the danger of being carried away by mathematical beauty into the fantasy land of modern physics in the form of Standard Quantum Mechanics, Standard Model and String Theory. Compare with Sabine Hossenfelder's Lost in Math and Max Tegmark's Mathematical Universe.

Let us recall the greatest successes of mathematical thinking about physics: 

• Newton's Theory of Gravitation                                    (N)
• Schrödinger's equation for the Hydrogen Atom.          (S)
• Maxwell's equations for electromagnetics.                   (M)

Newton's Theory of Gravitation is mathematics because it is captured in the following equations

•  $\Delta\phi =\rho$                                                         (N1)
• $F=-\nabla\phi$                                                         (N2)

where $\rho$ is mass density, $\phi$ is gravitational potential and $F$  is gravitational force all depending on a Euclidean space coordinate $x$ with corresponding Laplacian $\Delta$ and gradient $\nabla$ differential operator. 

These equations can be derived mathematically from a principle of conservation of energy and force. A point mass at $x=0$ then comes with a gravitational potential $-\frac{1}{\vert x\vert}$ and gravitational force scaling with $\frac{1}{\vert x\vert^2}$ as Newton's power two law. 

Newton's Model (N1) + (N2) capturing all of celestial mechanics through the differential operators $\Delta$ and $\nabla$, is the most formidable success of mathematical thinking all times. There was simply no  alternative for a celestial Creator, which allowed humans to get insight into the creation process by pure mathematical thinking. Amazing, right?  

Schrödinger's equation formulated in 1926 describes the charge density $\psi$ of a Hydrogen atom as the minimiser of the total energy $E_{tot}=E_{kin}+E_{pot}$ with 

• $E_{kin}=\frac{1}{2}\int\vert\nabla\psi\vert^2dx$
• $E_{pot}= -\int\frac{\psi^2}{\vert x\vert}dx$ 

under the side condition 

• $\int\psi^2dx =1$, 

as the solution of the eigenvalue problem 

• $-\frac{1}{2}\Delta\psi +V\psi = E\psi$                                  (S)

where $V(x)=-\frac{1}{\vert x\vert}$ is the electric Coulomb potential of the proton kernel, and $E$ is an eigenvalue. The connection between Coulomb potential and charge density is the same as between gravitational potential and mass density.

This model is not derived from basic principles like Newton's model, but is itself a first principle.

The success is that its eigenvalues match the observed spectrum of the Hydrogen atom to high precision, with a smallest eigenvalue $-\frac{1}{2}$ as the minimum of  $E_{tot}$ attained by a charge distribution finding the optimal combination of 

• being compressed around the kernel making $E_{pot}$ small
• paying a compression cost of $E_{kin}$

and so taking the simple form 

• $\psi (x) = \exp (-\vert x\vert )$

as a density decaying exponentially with the distance the kernel. 

Schrödinger's model of the ground state of a Hydrogen atom with a proton kernel surrounded by an electron charge density, can be compared with Newton's model of a Sun surrounded by an orbiting planet finding a balance of kinetic energy of motion and gravitational potential energy. 

The "compression energy" $E_{kin}$ of the electron then corresponds to the kinetic energy of the planet scaling with $\frac{1}{2}p^2$ with $p$ momentum . This gives a formal connection between $\nabla$ and $p$ as motivation of the name kinetic energy given to $E_{kin}$. 

The electron of a Hydrogen atom cannot be a point particle orbiting the kernel like a planet orbiting a sun, because a moving electron would radiate energy and so collapse into the kernel. The electron thus must have extension is space and it makes sense to associate $\nabla\psi$ with some form of "compression" or "strain" as in an elastic body. 

It is thus possible to argue that (S) is a most natural model of a Hydrogen atom as a kernel surrounded by an electron charge density finding an optimal solution to small potential energy at a compression cost. One can argue that the Creator had no choice when complementing celestial mechanics with Hydrogen atoms. 

Sum up so far: The macroscopic world of mechanics captured by (N) and the microscopic world of the Hydrogen atom captured by (S) as the result of pure mathematical thinking without need of observational input (Kant a priori), is a formidable success. Add to that (M).

But the success is not complete as concerns Schrödinger's equation since only the Hydrogen atom is covered. What then about a Schrödinger equation for atoms or molecules with many electrons? 

This was the question confronting Schrödinger and the world of modern physics in 1926 and the route that was taken has come to form modern physics all the way into our time. The idea was again to rely on pure mathematical thinking and so generalise (S) formally from one to many electrons simply by adding a new spatial coordinate for each new electron to arrive at a Schrödinger equation for an atom/molecule with $N$ electrons taking form in $3N$ spatial dimensions. No physics was involved in the generalisation, only mathematical formality. The mathematician von Neumann took control with his Mathematical Foundations of Quantum Mechanics 1932 leading physics into a strange universe of wave functions evolving in Hilbert spaces according to von Neumann Postulates (without physics), and physicists had no choice but to follow. 

The resulting Schrödinger equation was a multi-dimensional monster which did not describe any physics and in addition showed to be uncomputable except for very small $N$. The equation was easy to write down on a piece of paper just increasing the number of spatial dimensions, but real physics in three space dimension was present only in the case $N=1$. To save the situation the model was given a statistical interpretation by Max Born as all possibilities instead of particular realities as Standard Quantum Mechanics StdQM. Schrödinger never accepted StdQM.

RealQM offers a different generalisation in the original spirit of Schrödinger in the form of non-overlapping electron charge densities interacting by Coulomb potentials each coming with compression cost in total energy minimum of ground state. RealQM is a deterministic model of physics in three space dimensions and as such readily computable. 

Final sum up: (N) and (S) represent astounding complete successes of pure mathematical thinking, while StdQM appears as a monumental failure of too much belief in power of mathematical formality. RealQM gives hope to continued success of more modest mathematical thinking. The idea of a mathematical universe is great but should be combined with modesty according to the Greeks.

Paul Dirac claimed that very advanced mathematics was required to design the universeas, maybe as an excuse that his equation for the electron was very complicated beyond human understanding leading to Dirac's famous "Shut up and calculate". 

I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.  (von Neumann)