fredag 9 maj 2014

Why Insist on Quantum Mechanics Based on Magic and Contradiction?

The ground state of Helium is postulated to be $1s^2$ with two overlaying electrons with opposite spin and identical spherically symmetric spatial wave-functions in the first shell, which is not the ground state because its energy is too large. This is the starting point for the Schrödinger equation for many-electron atoms.

Here is a further motivation why it may be of interest to consider wave-functions for an atom with $N$ electrons as a sum of $N$ functions $\psi_1(x)$,…,$\psi_N(x)$, all depending on a common three-dimensional space coordinate $x$ (plus time)  as suggested in a previous post:
  • $\psi (x)=\psi_1(x)+\psi_2(x)+…+\psi_N(x)$.
We recall that Schrödinger's equation for the Hydrogen atom as the basis of quantum mechanics, takes the form:
  • $ih\frac{\partial\psi}{\partial t}=-\frac{h^2}{2m}\Delta\psi +V\psi$ for all $x$ and $t>0$,
with kernel potential $V(x)=-\frac{1}{\vert x\vert}$, $x$ a three-dimensional space coordinate, $t>0$ time, $h$ Planck's constant, $m$ the mass of an electron and corresponding one-electron wave-function $\psi (x,t)$ as solution. This equation is magically pulled out of a hat from the relation
  • $E =\frac{p^2}{2m} + V(x)$
expressing conservation of energy $E$ of a body of mass $m$ with position $x(t)$ moving in a potential $V(x)$ with momentum $p=m\frac{dx}{dt}$, by the following formal substitutions:
  • $E\rightarrow ih\frac{\partial}{\partial t}$,
  • $p\rightarrow\frac{h}{i}\nabla$,
followed by formal multiplication by $\psi$. Energy conservation for the Hydrogen atom then takes the form:
  • $E=K(t)+P(t)$ for all $t>0$, where
  • $K(t) =\frac{h^2}{2m}\int\vert\nabla\psi (x,t)\vert^2\, dx$ is the kinetic energy, 
  • $P(t)=\int \frac{\vert\psi (x, t)\vert^2}{\vert x\vert}dx$ is the potential energy   
of the electron, under the normalization
  • $\int\vert\psi (x,t)\vert^2\, dx=1$.
So far so good: The different energy levels $E$ of time-periodic solutions to Schrödinger's equation give the observed spectrum of the Hydrogen atom with corresponding wave-functions describing the distribution of the electron around the kernel. We see that the Laplace term gives rise to the kinetic energy as an effect of gradient regularization.  

But consider now the accepted standard text-book generalization of Schrödinger's equation to an atom with $N$ electrons:
  • $ih\frac{\partial\psi}{\partial t}=-\sum_{j=1}^N(\frac{h^2}{2m}\Delta_j -\frac{N}{\vert x_j\vert})\psi + \sum_{k < j}\frac{1}{\vert x_j-x_k\vert}\psi$,  
where $\psi (x_1,…,x_N,t)$ depends on $N$ three-dimensional space coordinates $x_1,…, x_N$ and time $t$, and $\Delta_j$ is the Laplace operator with respect to coordinate $x_j$, under the normalization
  • $\int\vert\psi\vert^2\, dx_1….dx_N=1$.
We see the appearance of the one-electron operators
  • $\frac{h^2}{2m}\Delta_j - \frac{N}{\vert x_j\vert}$     
with corresponding one-electron kinetic energies:
  • $K_j(t) =\frac{h^2}{2m}\int\vert\nabla_j\psi\vert^2\, dx_1…dx_N$, 
and electron-electron repulsion expressed by the coupling potential
  • $\sum_{k < j}\frac{1}{\vert x_j-x_k\vert}$.
We see that in this model each electron $j$ is equipped with its own three-dimensional space with coordinate $x_j$ and its own kinetic energy $K_j$, with interaction between the electrons only through the coupling potential.

The electron individuality and high dimensionality of the wave function $\psi (x_1,…x_N)$ is reduced by restriction to wave functions as products $\psi_1(x_1)…\psi (x_N)$ built from three-dimensional wave functions $\psi_1,…,\psi_N$ combined with symmetry or antisymmetry under permutations of the coordinates $x_1,…,x_N$, which eliminates all individuality of the electrons.

Extreme electron individuality is thus countered by permutations removing all individuality, but the individual one-electron kinetic energies $K_j$ are kept as if each electron keeps its individuality. This is strange.

To see the result, recall that the ground state of minimal energy of Helium with two electrons is supposed to be given by a symmetric wave function $\psi (x_1,x_2)$
  • $\psi (x_1,x_2)=\phi (x_1)\phi (x_2)$,      
where $\phi (x_1)\sim \exp(-2\vert x_1\vert )$ is spherically symmetric, the same for both electrons. The two electrons of the ground state of Helium are thus supposed to have identical spherically symmetric distributions denoted $1s^2$, see the periodic table above. The trouble is now that this configuration has energy (in Hartree units) $- 2.75$ while the observed energy is $-2.903$.

The true ground state is thus different from $1s^2$ and to handle this situation, while insisting that ground state still is $1s^2$ as in the table above, a so-called corrective perturbation is made introducing a dependence of $\psi (x_1,x_2)$ on $\vert x_1-x_2\vert$ in a Rayleigh-Ritz minimization procedure. This way a better correspondence with observation is reached, because separation of the electrons is now possible: If one electron is on one side of the kernel then the other electron is on the other side. But the standard message is contradictory:
  • The ground state configuration for Helium is $1s^2$, which however is not the ground state because its energy is too large ($-2.75$ instead of $-2.903$).
  • Smaller energy can be obtained by a perturbation computation but the corresponding electron configuration is hidden to readers of physics books, because the ground state is still postulated to be $1s^2$.     
If we minimize energy over wave functions of product form
  • $\psi (x_1,x_2)=\psi_1(x_1)\psi_2(x_2)$, 
without asking for symmetry, we find that the minimum is achieved with spherically symmetric  $\psi_1=\psi_2$, with too large energy as just noted. However, if we instead compute the kinetic energy based on the sum with common space space coordinate $x$
  • $\psi_1(x) +\psi_2(x)$ 
as suggested in the previous post, then separation of the electrons is advantageous allowing discontinuous electron distributions (joining smoothly) without cost of kinetic energy and better correspondence with observation is achieved.

  • The standard attribution of individual kinetic energy appears to make the individual electron distributions "too stiff" and thus favors overlaying electrons rather than separated electrons, requiring Pauli's exclusion principle to prevent overlaying of more than two electrons. 
  • If kinetic energy is instead computed from the sum of individual electron distributions, electron "stiffness" is reduced and separation favored. 
  • Since the standard individual one-electron attribution of kinetic energy is ad hoc,  there is little  reason to insist that kinetic energy must be computed this way, in particular when it leads to an incorrect ground state already for Helium. 
  • Attributing kinetic energy to a sum of electron wave-functions allows discontinuous electron distributions joining smoothly without cost of kinetic energy. Electron individuality is here kept as individual distribution in space, while kinetic energy is collectively computed from the assembly.  This would be the way to handle individuality in a collective macroscopic setting and there is no reason why this would not be operational also for microscopics.
  • Since the stated ground state as $1s^2$ for Helium is incorrect, there is no reason to believe that any of the other ground states listed in the standard periodic table is correct. 
  • If so, then the claim that the standard Schrödinger's equation explains the periodic table has little reason.
PS1 The standard argument is that the standard multi-d Schrödinger equation must be correct since there is no case known for which the multi-d wave-function solution does not agree exactly with what is observed! But this is not a correct argument, because (i) the multi-d Schrödinger equation cannot be solved, (ii) even if the wave-function could be determined its physical meaning is unclear and so comparison with reality is impossible. The standard argument is to turn (i) and (ii) from scientific disaster into monumental success by claiming that since the wave-function is impossible to determine, there is no way to prove that it is not correct.

Realizing that arguing this way does not follow basic scientific principle may open to searching for different forms of Schrödinger's equation, as non-linear systems of equations in three space dimensions instead of linear multi-d scalar equations, which are computable and have physical meaning, as suggested.

PS2 The standard way to handle that the standard linear multi-d Schrödinger equation is uncomputable is using Density Functional Theory (DFT) awarded the 1998 Nobel Prize in Chemistry, as a non-linear 3d scalar system in electron density. DFT results from averaging in the standard linear multi-d Schrödinger equation producing exchange correlation potentials which are impossible to determine. If the standard multi-d linear Schrödinger equation is questionable, then so is DFT.        

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