stdQM

In stdQM the ground state of an atom of kernel charge Z and N electrons, is represented by an anti-symmetric (taking an additional spin variable into account) wave function \psi (x) depending on a 3N-dimensional space variable x=(x_1,x_2,...,x_N) with the x_i independent 3d variables, which minimizes the total energy

  • TE(\psi )\equiv K(\psi )+PK(\psi )+PE(\psi ),

where now with \nabla_i the gradient with respect to x_i and dx=dx_1dx_2....dx_N,

  • K(\psi )=\sum_{i=1}^N\int_{\Re^{3N}}\frac{1}{2} \vert\nabla_i\psi (x)\vert^2\, dx
  • PK(\psi )=-\sum_{i=1}^N\int_{\Re^{3N}}\frac{Z}{\vert x_i\vert}\psi^2(x)\, dx
  • PE(\psi )=\sum_{i\neq j}\int_{\Re^{3N}}\frac{\psi^2(x)}{2\vert x_i-x_j\vert}dx

subject to the normalization

  • \int_{\Re^{3N}}\psi^2(x)\, dx = 1.

The equation expressing stationarity of total energy is the standard multi-d Schrödinger equation H\psi =E\psi with E the energy and H the standard multi-d Hamiltonian defined by

  • H = -\sum_{i=1}^N\frac{1}{2}\Delta_i-\sum_{i=1}^N\frac{Z}{\vert x_i\vert}+\sum_{i\neq j}\frac{1}{2\vert x_i-x_j\vert}

and \Delta_i the Laplacian with respect to x_i, as a partial differential equation for a wave function depending on 3N spatial variables. As such the wave function can be given a physical meaning in physical 3d space only in the case of Hydrogen with N=1. For N\ge 2 the wave function is given a non-physical probabilistic meaning as a probability distribution of electron configurations with electron j at position x_j for j=1,...,N. We see that the standard Schr\”odinger equation is a linear multi-d partial differential equation.

The assumption of antisymmetry of the wave function guarantees satisfaction of Pauli’s Exclusion Principle stating that two electrons (with the same spin) cannot be at the same place (while two electrons with opposite spin can). Antisymmetry means that \psi (x) changes sign upon permutation of any two electron positions x_j and x_k, from which follows that \psi (x)=0 if x_j=x_k, which is interpreted to mean that the probability of electron j and k being at the same place is zero.

Schrödinger’s equation for an atom with 10 electrons thus involves 30 independent spatial variables/coordinates (up to anti-symmetry), and with a modest resolution of 100 mesh points in each coordinate direction, the total number of mesh points would be 100^{30}=10^{60} which is more than the estimated total number of atoms in the Universe. The multi-d Schr\”odinger equation for an atom with several electrons is thus uncomputable and solutions are thus unknowable and as such of questionable scientific value.

We understand that the unphysical meaning of the standard multi-d wave function goes hand in hand with uncomputability. What is needed is a wave function which both has physical meaning and is computable, while the wave function offered by standard quantum mechanics is both unphysical and uncomputable. The effect is a quantum mechanics of modern physics where causality and physicality no longer have any meaning and electrons are assumed to be present both everywhere and nowhere.

stdQM offers a linear high-dimensional uncomputable model, which is turned computable in Hartree-Fock and DFT at the expense of (more or less severe) distortion. On the other hand realQM offers a 3d non-linear model, which we show to be computable in laptop computations of ground state energy of atoms in the periodic table in spherical symmetry in agreement with reference values. This indicates that realQM ab initio computations may be feasible in large generality.

The linearity of std Schrödinger is a cornerstone of stdQM, which has led modern physicists to believe that quantum computers can be constructed based on superposition of atomic states, so far however only fantasy. But the rationale of insisting to model atom physics with a linear equation is lacking. Macro-physics is not linear and there is no reason to believe that micro-physics is. To believe in superposition of life and death of a Schrödinger cat is superstition physics.

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