## tisdag 21 juni 2016

### New Quantum Mechanics 3: Why?

Modern physics is being based on (i) relativity theory and (ii) quantum mechanics, both viewed to be correct beyond any conceivable doubt, but nevertheless (unfortunately) being incompatible. The result is a modern physics based on shaky grounds of contradictory theories from which anything can emerge, and so has done in the form of string theory and multiversa beyond thinkable experimental verification.

The basic model of quantum mechanics is Schrödinger's equation as a linear equation in a wave function depending on $3N$ spatial dimensions for an atom with $N$ electrons. Schrödinger's equation is an ad hoc model arrived at by a purely formal extension of classical mechanics without direct physical meaning and rationale. Schrödinger's equation is thus viewed as being given by God with the job of physical interpretation being left to humanity in endless quarrels. In this sense quantum mechanics is rather religion than science and the present state of physics maybe a fully logical result.

Experimental support for Schrödinger's equation is in incontestable form only available in the case of Hydrogen with $N=1$, since for larger $N$ the multidimensionality prevents both analytical and computational solution.  The message of books on quantum mechanics that solutions of Schrödinger's equation always (have to) agree with observations, rather reflect a belief that a God-given equation cannot be wrong, than actual human experience.

But if we as scientists do not welcome the idea of an equation given by God beyond human comprehension,  then we may find motivation to search for an alternative atomic model which is computable and thus is possible to compare with physical experiment.  This is my motivation anyway.

And God said:

And then there were Atoms!

### New Quantum Mechanics 2: Computational Results

I have now tested the atomic model for an atom with $N$ electrons of the previous post formulated as a classical free boundary problem in $N$ single-electron charge densities with non-overlapping supports filling 3d space with joint charge density as a sum of electron densities being continuously differentiable across inter-electron boundaries.

I have computed in spherical symmetry on an increasing sequence of radii dividing 3d space into a sequence of shells filled by collections of electrons smeared into spherically symmetric shell charge distribution. The electron-electron repulsive energy is computed with a reduction factor of $\frac{n-1}{n}$ for the electrons in a shell with $n$ electrons to account for lack of self repulsion.

Below is a typical result for Xenon with 54 electrons organised in shells with 2, 8, 18, 18 and 8 electrons with ground state energy -7413 to be compared with -7232 measured and with the energy distribution in the 5 shells displayed in the order of total energy, kinetic energy, kernel potential energy and inter-electron energy. Here the blue curve represents electron charge density, green is kernel potential and red is inter-electron potential. The inter-shell boundaries are adaptively computed  so as to represent a preset 2-8-18-18-8 configuration in iterative relaxation towards a ground state of minimal energy.

In general computed ground state energies agree with measured energies within a few percent for all atoms up to Radon with 86 electrons.

The computations indicate that it may well be possible to build an atomic model based on non-overlapping electronic charge densities as a classical continuum mechanical model with electrons keeping individuality by occupying different regions of space, which agrees reasonably well with observations. The model is an $N$-species free boundary problem in three space dimensions and as such is readily computable for any number of $N$ for both ground states, excited states and dynamic transitions between states.

We recall the the standard model in the form of Schrödinger's equation for a wave function depending on $3N$ space dimensions, is computationally demanding already for $N=2$ and completely beyond reach for larger $N$. As a result the full $3N$-dimensional Schrödinger equation is always replaced by some radically reduced model such as Hartree-Fock with optimization over a "clever choice" of a few "atomic orbitals", or Thomas-Fermi and Density Functional Theory with different forms of electron densities.

The present model is an electron density model, which as a free boundary problem with electric individuality is different from Thomas-Fermi and DFT.

We further recall that the standard Schrödinger equation is an ad hoc model with only formal justification as a physical model, in particular concerning the kinetic energy and the time dependence, and as such should perhaps better not be taken as a given ready-made model which is perfect and as such canonical (as is the standard view).

Since this standard model is uncomputable, it is impossible to show that the results from the model agree with observations, and thus claims of perfection made in books on quantum mechanics rather represent an ad hoc preconceived idea of unquestionable ultimate perfection than true experience.

## onsdag 1 juni 2016

### New Quantum Mechanics 1 as Classical Free Boundary Problem

Let me (as a continuation of the sequence of posts on Finite Element Quantum Mechanics 1-5) present an alternative formulation of the eigenvalue problem for Schrödinger's equation for an atom with $N$ electrons starting from an Ansatz for the wave function
• $\psi (x) = \sum_{j=1}^N\psi_j(x)$      (1)
as a sum of $N$ electronic real-valued wave functions $\psi_j(x)$, depending on a common 3d space coordinate $x\in R^3$ with non-overlapping spatial supports $\Omega_1$,...,$\Omega_N$, filling 3d space, satisfying
• $H\psi = E\psi$ in $R^3$,       (2)
where $E$ is an eigenvalue of the (normalised) Hamiltonian $H$ given by
• $H(x) = -\frac{1}{2}\Delta - \frac{N}{\vert x\vert}+\sum_{k\neq j}V_k(x)$ for $x\in\Omega_j$,
where $V_k(x)$ is the potential corresponding to electron $k$ defined by
• $V_k(x)=\int\frac{\psi_k^2(y)}{2\vert x-y\vert}dy$, for $x\in R^3$,
and the wave functions are normalised to correspond to unit charge of each electron:
• $\int_{\Omega_j}\psi_j^2(x) dx=1$ for $j=1,..,N$.
One can view (2) as a formulation of the eigenvalue problem for Schrödinger's equation, starting from an Ansatz for the total wave function as a sum of electronic wave function according to (1), as a classical free boundary problem in $R^3$, where the electron configuration is represented by a partition of $R^3$ into non-overlapping domains representing the supports of the electronic wave functions $\psi_j$ and the total wave function $\psi$ is continuously differentiable.

Defining $\rho_j = \psi_j^2$, we have
• $\psi\Delta\psi = \frac{1}{2}\Delta\rho-\frac{1}{4\rho}\vert\nabla\rho\vert^2$,
and thus (2) upon multiplication by $\psi$ takes the form
• $-\frac{1}{4}\Delta\rho+\frac{1}{8\rho}\vert\nabla\rho\vert^2-\frac{N\rho}{\vert x\vert}+V\rho = E\rho$ in $R^3$,                   (3)
where
• $\rho_j\ge 0$, $support(\rho_j)=\Omega_j$ and $\rho_j=0$ else,
• $\int_{\Omega_j}\rho_jdx =1$,
• $\rho =\sum_j\rho_j$,
• $V\rho=\sum_{k\neq j}V_k\rho_j$ in $\Omega_j$,
• $\Delta V_j=2\pi\rho_j$  in $R^3$.
The model (3) (or equivalently (2)) is computable as a system in 3d and will be tested against observations. In particular the ground state of smallest eigenvalue/energy E is computable by parabolic relaxation of (3) in $\rho$. Continuity of $\psi$ then corresponds to continuity of $\rho$.

We can view the formulation (3) in the same way as that explored for gravitation, with the potential $V_j$ primordial and the electronic density $\rho_j$ defined by $\rho_j =\frac{1}{2\pi}\Delta V_j$ as a derived quantity, with in particular total electron-electron repulsion energy given by the neat formula
• $\frac{1}{2\pi}\sum_{k\neq j}\int V_k\Delta V_jdx=-\frac{1}{2\pi}\sum_{k\neq j}\int\nabla V_k\cdot\nabla V_jdx$
in terms of potentials with an analogous expression for the kernel-electron attraction energy.

For the choice of free boundary condition see later post.

### Many Big Bangs: Universe Bigger Than You Think

Astronomer Royal Lord Rees has made a statement:
• There may have been more than one Big Bang, the Astronomer Royal has said and claims the world could be on the brink of a revolution as profound as Copernicus discovering the Earth revolved around the Sun.
• Many people suspect that our Big Bang was not the only one, but there’s a whole ensemble of Big Bangs, a whole archipelago of Big Bangs.
• The theory is still highly controversial, but Lord Rees said he would ‘bet his dog’ on the theory being true.
This fits with the view I have presented in posts on a new view on gravitation, dark matter and dark energy, with gravitational potential $\phi$ viewed as primordial from which matter density $\rho$, which may be both positive and negative, is generated by
• $\rho = \Delta\phi$,
through local action in space of the Laplacian $\Delta$. In this model a Big Bang corresponds to a small local fluctuation of $\phi$ around zero, which generates an much bigger fluctuation of matter density by the action of the Laplacian.

In this model substantial matter may be generated locally from small fluctuations of gravitational potential opening the possibility of an endless number of Big Bangs seemingly created out of nothing.

You can test the model in the app Dark Energy on App Store.