måndag 19 december 2016

New Quantum Mechanics 21: Micro as Macro

The new quantum mechanics as realQM explored in this sequence of posts offers a model for the microscopic physics of atoms which is of the same form as the classical continuum mechanical models of macroscopic physics such as Maxwell's equations for electro-magnetics, Navier's equations for solid mechanics and Navier-Stokes equations for fluid mechanics in terms of deterministic field variables depending on a common 3d space coordinate and time.

realQM thus describes an atom with $N$ electrons realQM as a nonlinear system of partial differential equations in $N$ electronic wave functions depending on a common 3d space coordinate and time.

On the other hand, the standard model of quantum mechanics, referred to as stdQM, is Schrödinger's equation as a linear partial differential equation for a probabilistic wave function in $3N$ spatial coordinates and time for an atom with $N$ electrons.  

With realQM the mathematical models for macroscopic and microscopic physics thus have the same form and the understanding of physics can then take the same form. Microphysics can then be understood to the same extent as macrophysics.

On the other hand, the understanding of microphysics according to stdQM is viewed to be fundamentally different from that of macroscopic physics, which effectively means that stdQM is not understood at all, as acknowledged by all prominent physicists.

As an example of the confusion on difference, consider what is commonly viewed to be a basic property of stdQM, namely that there is limit to the accuracy that both position and velocity can be determined on atomic scales, as expressed in Heisenberg's Uncertainty Principle (HUP).

This feature of stdQM is compared with the situation in macroscopic physics, where the claim is that both position and velocity can be determined to arbitrary precision, thus making the case that microphysics and microphysics are fundamentally different.

But the position of a macroscopic body cannot be precisely determined by one point coordinate, since  a macroscopic body is extended in space and thus occupies many points in space.  No one single point determines the position of and extended body. There is thus also a Macroscopic Uncertainty Principle (MUP).

The argument is then that if the macroscopic body is a pointlike particle,  then both its position and velocity can have precise values and thus there is no MUP. But a pointlike body is not a macroscopic body and so the argument lacks logic.

The idea supported by stdQM that the microscopic world is so fundamentally different from the macroscopic world that it can never be understood, thus may well lack logic. If so that could open to understanding of microscopic physics for human beings with experience from macroscopic physics.

If you think that there is little need of making sense of stdQM, recall Feynman's testimony:
  • We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I’m an old enough man that I haven’t got to the point that this stuff is obvious to me. Okay, I still get nervous with it ... You know how it always is: every new idea, it takes a generation or two until it becomes obvious that there’s no real problem. I cannot define the real problem, therefore I suspect that there is no real problem, but I’m not sure there’s no real problem. (Int. J. Theoret. Phys. 21, 471 (1982).) 
It is total confusion, if it is totally unclear if there is a problem or no problem and it is totally clear that nobody understands stdQM....

Recall that stdQM is based on a linear multi-dimensional Schrödinger equation, which is simply picked from the sky using black magic ad hoc formalism, which could be anything, and is then taken as a revelation about real physics when interpreted by reversing the black magics. 

This is like scribbling down a sign/equation at random without intentional meaning, and then giving the sign/equation an interpretation as if it had an original meaning, which may well be meaningless, instead of expressing a meaning in a sign/equation to discover consequences and deeper meaning.   

fredag 16 december 2016

New Quantum Mechanics 20: Shell Structure

Further computational exploration of realQM supports the following electronic shell structure of an atom:

Electrons are partitioned into an increasing sequence of main spherical shells $S_1$, $S_2$,..,$S_M$ with each main shell $S_m$ subdivided into two half-spherical shells each of which for $m>2$ is divided into two angular directions into $m\times m$ electron domains thus with a total of $2m^2$ electrons in each full shell $S_m$.  The case $m=2$ is special with the main shell divided radially into two subshells which are each divided into half-spherical subshells each of which is finally divided azimuthally, into $2\times 2$ electron domains for $S_2$ subshell, thus with a total of $2m^2$ electrons in each main shell $S_m$ when fully filled, for $m=1,...,M$, see figs below.

This gives the familiar sequence 2, 8, 18, 32,.. as the number of electrons in each main shell.

4 subshell of S_2
8 shell as variant of full S_2 shell
 9=3x3 halfshell of S_3

The electron structure can thus be described as follows with parenthesis around main shells and radial subshell partition within parenthesis:
  • (2)+(4+4)
  • (2)+(4+4)+(2)
  • ...
  • (2)+(4+4)+(4+4) 
  • (2)+(4+4)+(8)+(2)
  • ....
  • (2)+(4+4)+(18)+(2)
  • ...
  • (2)+(4+4)+(18)+(8)
Below we show computed ground state energies assuming full spherical symmetry with a radial resolution of 1000 mesh points, where the electrons in each subshell are homogenised azimuthally, with the electron subshell structure indicated and table values in parenthesis. Notice that the 8 main shell structure is repeated so that in particular Argon with 18 electrons has the form 2+(4+4)+(4+4):

Lithium (2)+1: -7.55 (-7.48)                  1st ionisation:      (0.2)
Beryllium (2)+(2): -15.14 (-14.57)           1st ionisation: 0.5 (0.35)
Boron (2)+(2+1): -25.3 (-24.53)               1st ionisation: 0.2 (0.3)
Carbon (2)+(2+2): -38.2  (-37.7)               1st ionisation 0.5 (0.4)
Nitrogen (2)+(3+2):  -55.3 (-54.4)            1st ionisation  0.5  (0.5)
Oxygen (2)+(3+3): -75.5 (-74.8)               1st ionisation  0.5  (0.5)
Fluorine (2)+(3+4):  -99.9   (-99.5)            1st ionisation  0.5      (0.65)
Neon (2)+(4+4):   -132.4     (-128.5  )        1st ionisation 0.6        (0.8)
Sodium (2)+(4+4)+(1): -165 (-162)
Magnesium (2)+(4+4)+(2): -202  (-200)
Aluminium (2)+(4+4)+(2+1): -244 (-243)
Silicon (2)+(4+4)+(2+2): -291 (-290)
Phosphorus (2)+(4+4)+(3+2): -340 (-340)
Sulphur (2)+(4+4)+(4+2): -397 (-399)
Chlorine (2)+(4+4)+(3+4): -457 (-461)
Argon: (2)+(4+4)+(4+4): -523 (-526)
Calcium: (2)+(4+4)+(8)+(2): -670 (-680)
Titanium: (2)+(4+4)+(10)+(2): -848 (-853)
Chromium: (2)+(4+4)+(12)+(2): -1039 (-1050)
Iron: (2)+(4+4)+(14)+(2): -1260 (-1272)
Nickel: (2)+(4+4)+(16)+(2): -1516 (-1520)
Zinc: (2)+(4+4)+(18)+(2): -1773 (-1795)
Germanium: (2)+(4+4)+(18)+(2+2): -2089 (-2097)
Selenium: (2)+(4+4)+(18)+(4+2):- 2416 (-2428)
Krypton: (2)+(4+4)+(18)+(4+4): -2766 (-2788)
Xenon: (2)+(4+4)+(18)+(18)+(4+4): -7355  (-7438)
Radon: (2)+(4+4)+(18)+(32)+(18)+(4+4): -22800 (-23560)

We see good agreement even with the crude approximation of azimuthal homogenisation used in the computations.

To see the effect of the subshell structure we compare Neon: (2)+(4+4) with Neon: (2)+(8) without the (4+4) subshell structure, which has a ground state energy of -153, which is much smaller than the observed -128.5.  We conclude that somehow the (4+4) subdivision of the second is preferred before a subdivision without subshells. The difference between (8) and (4+4) is the homogeneous Neumann condition acting between subshells, tending to increase the width of the shell and thus increase the energy.

The deeper reason for this preference remains to describe, but the intuition suggests that it relates to the shape or size of the domain occupied by an electron.  With subshells electron domains are obtained by subdivision in both radial and azimuthal direction, while without subshells there is only azimuthal/angular subdivision of each shell.

We observe that ionisation energies, which are of similar size in different shells, become increasingly small as compared to ground state energies, and thus are delicate to compute as the difference between the ground state energies of atom and ion.

Here are sample outputs for Boron and Magnesium as functions of distance $r$ from the kernel along the horizontal axis :

We observe that the red curve depicting shell charge $\psi^2(r)r^2dr$ per shell radius increment $dr$, is roughly constant in radius $r$, as a possible emergent design principle. More precisely, $\psi (r)\sim \sqrt{Z}/r$ mathches with $d_m\sim m^2/Z$ and $r_m\sim m^3/Z$ with $d_m$ the width of shell $S_m$ and thus the width of the subshells of $S_m$ scaling with $m/Z$, and thus the width of electrons in $S_m$ scaling with $m/Z$.

We thus have $\sum_mm^2\sim M^3\sim Z$ and with $d_m\sim m^2/Z$ the atomic radius $\sum_md_m\sim M^3/Z\sim 1$ is basically the same for all atoms, in accordance with observation.

Further, the kernel potential energy and thus the total energy in $S_m$ scales with $Z^2/m$ and the total energy by summation over shells scales with $\log(M)Z^2\sim \log(Z)Z^2$, in close correspondence with $Z^{\frac{1}{3}}Z^2$ by density functional theory.

Recall that the electron configuration of stdQM is based on the eigen-functions for Schrödinger's equation for the Hydrogen atom with one electron, while as we have seen that of realQM rather relates to spatial partitioning. Of course, eigen-functions express some form of partitioning, and so there is a connection, but the basic problem may concern partitioning of many electrons rather than eigen-functions for one electron.


torsdag 8 december 2016

Quantum Mechanics as Theory Still Without Meaning

Yet another poll (with earlier polls in references) shows that physicists still today after 100 years of deep thinking and fierce debate show little agreement about the stature of quantum mechanics as the prime scientific advancement of modern physics.

The different polls indicate that less than 50% of all physicists today adhere to the Copenhagen Interpretation, as the main text book interpretation of quantum mechanics. This means that quantum mechanics today after 100 years of fruitless search for a common interpretation, remains a mystery without meaning. Theory without interpretation has no meaning and science without meaning cannot be real science.

If only 50% of physicists would agree on the meaning of the basic text book theories of classical physics embodied in Newton/Lagranges equations of motion, Navier's equation for solid mechanics, Navier-Stokes equations for fluid dynamics and Maxwell's equations for electromagnetic, that would signify a total collapse of classical physics as science and subject of academic study.

But this not so: classical physics is the role model of science because there is virtually no disagreement on the formulation and meaning of these basic equations.

But the polls show that there is no agreement on the role and meaning of Schrödinger's equation as the basis of quantum mechanics, and physicists do not seem to believe this will ever change. This is far from satisfactory from scientific point of view.

This is my motivation to search for a meaningful quantum mechanics in the form of realQM presented in recent posts. Of course you may say that for many reasons my chances of finding some meaning are very small, but science without meaning cannot be real science.

PS Lubos Motl, as a strong proponent of a textbook all-settled Copenhagen interpretation defined by himself, reacts to the polls with
  • The foundations of quantum mechanics were fully built in the 1920s, mostly in 1925 or at most 1926, and by 1930, all the universal rules of the theory took their present form...as the Copenhagen interpretation. If you subtract all these rules, all this "interpretation", you will be left with no physical theory whatsoever. At most, you will be left with some mathematics – but pure mathematics can say nothing about the world around us or our perceptions.
  • In virtually all questions, the more correct answers attracted visibly greater fractions of physicists than the wrong answers.
Lubos claims that more correct views, with the true correct views carried by only Lubos himself, gathers a greater fraction than less correct views, and so everything is ok from Lubos point of view. But is greater fraction sufficient from scientific point of view, as if scientific truth is to be decided by democratic voting? Shouldn't Lobos ask for 99.9% adherence to his one and only correct view? If physics is to keep its position as the king science?

Or is modern physics instead to be viewed as the root of modernity through a collapse of classical ideals of rationality, objectivity and causality?