måndag 27 april 2015

Physical Computable Quantum Mechanics: Towards The Periodic Table

The basic mathematical of quantum mechanics is Schrödinger's wave equation in terms of a wave function $\psi (x_1,x_2,...,x_N;t)$ depending on $N$ three-dimensional space coordinates $x_1$,...$x_N$ for an atom/ion with $N$ electrons, together with a time coordinate $t$. The interpretation of the wave function, thus depending on altogether $3N$ space dimensions plus time,  according to the so-called Copenhagen interpretation by Bohr-Heisenberg-Born, is that $\psi (x_1,x_2,...,x_N;t)$ is the probability distribution at time $t$ of finding electron $j$ at position $x_j$ for $j=1,...,N$.

Gerard t'Hooft, Nobel Prize in Physics 1999, summarizes the state of quantum mechanics 73 years after the formulation by Schrödinger of his equation, in the book In Search of the Ultimate Building Blocks:
  • Quantum mechanics works beautifully, there is little doubt about that.
  • However, a very peculiar question presented itself: what do these equations actually mean? What is it they are describing?
  • It seems as if electrons can "exist" at different places simultaneously.
  • The rules of quantum mechanics work so well that it has become very hard to refute them. Ingenious tricks discovered by Heisenberg, Dirac and many others further improved and streamlined the general rules.
  • But Einstein and Schrödinger always had serous objections to the Copenhagen Interpretation. Maybe it works all right, but where exactly is the electron, at the point $x$ or $y$, in reality
  • The history books say that Bohr has proved Einstein wrong. But others, including myself, suspect that, in the long run, the Einsteinian view might return.
  • Probabilities and statistics are mistreated a great deal, even by physicists. Some have uttered, for instance, the theory that all possibilities for certain events are being realized in "parallel worlds"....To my sober mind this is nonsense.
I agree with t'Hooft/Einstein/Schrödinger: Quantum mechanics may work if handled with proper "tricks", but the $3N$-dimensional wave function as a probability distribution cannot be the right thing to describe electron configurations. The reason is double: First, a probability distribution has no direct physical meaning and thus cannot describe the way electrons actually behave. Second, a $3N$ dimensional wave function is uncomputable as soon as $N$ is not very small like 2.  

The first "trick" used to get a computable wave function introduced by Hartree, was to make an Ansatz as a product of $N$ wave functions $\psi_j(x_j)$  each depending on a three-dimensional space coordinate (plus time):
  • $\psi (x_1,...,x_N)=\psi_1(x_1)\psi_2(x_2)...\psi_N(x_N),$ 
which reduces Schrödinger's $N$-electron wave equation in $3N$ space dimensions, to a system of one-electron wave equations in 3d. 

I have explored another "trick", which possibly can be elevated from trick to fundamental physical model, with the wave function instead expressed as a sum 
  • $\psi (x)=\psi_1(x)+\psi_2(x)+...\psi_N(x)$
in terms of $N$ functions $\psi_1(x),...,\psi_N(x)$ depending on a a common 3d space coordinate $x$ (plus time), with non-overlapping supports $\Omega_j$ filling 3d space. Each wave function $\psi_j(x)$ can then be interpreted as the "charge density" of electron $j$, following the original idea of Schrödinger, subject to the normalization
  • $\int_{\Omega_j}\vert\psi_j(x)\vert^2dx =  1.$
The time-independent ground state $\psi (x)$ is then determined as the continuous differentiable function $\psi (x)=\psi_1(x)+\psi_2(x)+...\psi_N(x)$ based on a non-overlapping partition of 3d space into subdomains $\Omega_j$ serving as the supports of the $\psi_j$, which minimizes the total energy functional
  • $\frac{1}{2}\int\vert\nabla\psi\vert^2 dx-\int\frac{N\psi^2}{\vert x\vert}dx+\sum_{k\neq j}\int_{\Omega_j}\int_{\Omega_k}\frac{\psi_j^2(x_j)\psi_k^2(x_k)}{2\vert x_j-x_k\vert}dx_jdx_k.$ 
This is a free-boundary electron (or charge) density formulation keeping the individuality of the electrons, which can be viewed as a "smoothed $N$-particle problem" of interacting non-overlapping "electron clouds" under Laplacian smoothing. Preliminary computation (see Quantum Contradictions and Physical Quantum Mechanics) with this model shows a surprisingly good agreement with observations. More precise computations will be reported shortly.

My dream is to rediscover the periodic table from this model. It is not impossible that the dream can come true.

For example, the ground state of Helium in this model consists of two disjoint (contacting) half-spherical electron clouds with energy in close agreement with observation.
This is to be compared with the postulated ground state according to the standard Schrödinger equation considered to be two overlaying spherical electron configurations named as $1s2$, which however has wrong energy and thus is not the true ground state, and thus asks for perturbative correction.

PS1 In the series Quantum Contradictions 1-29 we considered a model closely related to (1) with each one-electron wave function $\psi_j$ a smooth function defined in 3d space satisfying a one-electron wave equation with the Laplacian smoothing acting on the sum of the wave functions,  and not on each individual wave function, with the effect of reducing contributions to the smoothing energy from angular variation. In this case the one-electron wave functions have overlapping support, while being largely separated by repulsion and loosing individuality under overlapping Laplacian smoothing.

PS2  Computing an approximation of the ground-state energy $E$ of Helium using the two normalized separated half-spherical wave functions
  • $\psi_1(x)=\sqrt{\frac{2\alpha^3}{\pi}}\exp(-\alpha\vert x\vert )$ for $x=(x_1,x_2,x_3)$ with $x_3\ge 0$
  • $\psi_2(x)=\sqrt{\frac{2\alpha^3}{\pi}}\exp(-\alpha\vert x\vert )$ for $x$ with $x_3\le 0$,
 we find with $\alpha =27/16$ the following value:
  • $E=-2.902\pm 0.002$,
to be compared with the observed $-2.903$. The standard 1s2 ground state of two overlaying fully spherical wave functions gives a best value of -2.85 asking for so called perturbation correction, which effectively introduces electron separation. We have thus found evidence that the ground state of Helium is not the standard 1s2 state with two overlaying spherical wave functions, without or with perturbation correction, but instead consists of two non-overlapping half-spherical wave functions. If this conclusion indeed shows to be  correct, then electronic wave functions will have to be recomputed for all atoms. Mind-boggling!

PS3 For Li+ we get E = -7.32 (-7.28), for Be2+ we get = -13.72 (-13.65) and for B3+ we get -22.12 (-22.03) with measured value in parenthesis....more to come...

13 kommentarer:

  1. This reevaluation of the fundaments of atomic theory is truly revolutionary, but the physicists seem to be silent. I wonder why.

  2. The physicists are silent because as long as a theory only predicts a ground state energy of an atom, to some accuracy, there's no knowledge of transitivity to the type of systems that we're really interested in. Predicting ground state energies are not that impressing, Helium for instance can be determined from first principles to one part in 10^15 by using a correct theory beyond the schoolbook example of superimposed s-orbitals. That approximation though, suffices for most applications since there is no real merit in shuffling precision in decimal places that doesn't matter in the end. So as long as a model doesn't predict anything of real importance, no chemist or physicist would bother. If the model whould manage to correctly predict properties for ferroelectric materials for instance, then maybe there would be some interest...

  3. The goal of the new physical computable quantum mechanics is to model the radiating atom or molecule, for example, which is beyond what is possible with standard multidimensional Schrödinger.

  4. That's what you use the Dirac equation for...

  5. Or Schrodinger with minimal coupling for some applications...

  6. Yes, of course everything has already been done, long ago...that is why physicists today have nothing better to do than spend their days with string theory and parallel worlds...and the period table is of course readily explained by multidim Schrödinger equation, although it is uncomputable and solutions unknown...so why bother at all?

  7. You desperately need to educate yourself in what physicist does. The things you mention here, string theory and parallel worlds are a gross misrepresentation of what the overwhelming bulk of physicists actually do or really care about.

    The following blog post discuss the strange kind of misconceptions and misrepresentations that mainstream media (also you self as well here it seems) and popular science media have about what "most physicists" work on.


    The conclusion is that 2% is an estimated upper bound on the amount of physicists that even bother about issues like the ones you mention. But if it's 1%, 5% or 2% doesn't really matter, the amount of resources that little minority consumes is really a piss in the ocean compared to the total amount, so why bother?

  8. Yes, of course everything has already been done, long ago...that is why mathematicians have nothing better to do than spend their days with solving the Riemann Hypothesis...

    Is this a good representation of what you mathematicians do?

  9. Two issues regarding your total wavefunction.

    First, you express the total wavefunction as the sum of each individual wavefunction. But, with the normalization you use, the physical dimension of the wavefunctions is L^(-3/2). What kind of physical object is that? How can you meaningfully construct a sum of that quantity, given that you want to describe something physically real in the classical sense?

    Secondly, given your interpretation of the wavefunction, how do you interpret the solution for the free particle Schrödinger equation since that is a plane wave? And then given a scattering event for this wavefunction. The solution to the Schrödinger equation is then a wave scattering in all directions, how do you interpret that?

  10. I was hoping that you had answers to these questions.


  11. As I have said, the model I consider can be viewed as a smoothed particle model and as such has a direct physical meaning, including wave formation just as a string of beads can form a wave.