Quantum Illusions

There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks. (Schrödinger)

A physicist will tell you that the electron configuration of an atom or molecules is fully described by Standard Quantum Mechanics StdQM in terms of a wave function $\Psi (t)$ as solution to Schrödinger's equation with $H$ a Hamiltonian operator:

• $i\dot\Psi (t)  = H\Psi (t)$ for $t>0$,                (S)

where $\dot\Psi =\frac{\partial\Psi}{\partial t}$ which evolves from a given initial state $\Psi (0)$ to any later state $\Psi (t)$ for $t>0$. The objective is to predict the state $\Psi (t)$ for some $t>0$ from specification of $H$ and $\Psi (0)$, thus as solution to an initial value problem. The wave function $\Psi$ depends on $3N$ spatial coordinates for an atom or molecule with $N$ electrons, and so has no direct physical meaning, only a non-physical statistical meaning.  

Specification of $H$ requires information about positions of atomic kernels, which may change over time. Specification of $\Psi (0)$ requires information of the full $3N$-dimensional statistics of the initial state, which is not experimentally measurable. The only possibility is then to prepare the system to have a certain artificial initial state which can be described by analytical mathematics but then meets the difficulty of being possible to realise experimentally. 

The net result is that (S) as an initial value problem is not a real model of real physics, rather an illusion of a model or a model of an illusion. Speaking about solutions of (S) as describing the time-dependent physics of an atom or molecule thus makes little real sense. 

What remains to speak about is then time-independent solutions $\bar\Psi$ of the eigenvalue problem 

• $H\bar\Psi =E\bar \Psi$                                   (SE)

where $E$ is an eigenvalue as a real number and $\bar\Psi$ is an eigenfunction. In this case, no initial values enter, and specification of $H$ from static kernel positions is enough. The eigenvalues represent energies of eigenfunctions, which are measurable as lines in the spectrum of an atom. The corresponding time-dependent solution $\Psi (t)$ is given by the formula

• $\Psi (t)=\exp (-iEt)\bar\Psi$,                       

which gives meaning to (S) since the difference of eigenvalues appear as spectral lines.

We conclude that the eigenvalue problem (SE) can carry physical meaning, but the $3N$-dimensionality makes (SE) uncomputable for most systems, and so also largely represents an illusion. 

This is to be contrasted with the often repeated claim that StdQM is (i) the most successful of all theories about physics with (ii) no prediction in violation with experiment (with the caveat that few predictions are possible).  

We compare StdQM with RealQM based on a new Schrödinger equation in 3d for which both (S) and (SE) have direct physical meaning and are computable.