tisdag 8 oktober 2013

From Impossible Fiction to Possible Reality in Quantum Mechanics

The wave function $\Psi$ as a solution to Schrödinger's equation supposedly describing the quantum mechanics of an atom with $N$ electrons, depends on N three-dimensional spatial variables $x_1,..., x_N$ (and time), altogether $3N$ spatial dimensions, with $\vert \Psi (x_1,..., x_N)\vert^2$ interpreted as the probability of the configuration with electron $j$ located at position $x_j$.

The wave function $\Psi (x_1,...,x_N)$ is thus supposed to carry information about all possible electron configurations and as such contains an overwhelming amount of information, which however is not really accessible because of an overwhelming computational cost already for small N with difficulties starting already for $N = 2$.

To handle this difficulty drastic reductions in complexity are being made by seeking approximate solutions as wave functions with drastically restricted spatial variation based on heuristics. There are claims that this is feasible using structural properties of the wave function (but full scale computations seem to be missing).

An alternative approach would be to seek $N$ wave functions $\Psi_1,..., \Psi_N$, depending on a common three-dimensional space coordinate $x$, with $\Psi_j(x)$ carrying information about the presence of particle $j$, as a form of Smoothed Particle Mechanics (SPM) as a variant of classical particle mechanics. The corresponding Schrödinger equation consists of a coupled system of one-electron equations in the spirit of Hartree, and is readily computable even for N large.

The solution $\Psi_1(x),..., \Psi_N(x)$, of the system would give precise information about one possible electron configuration. If this is a representative configuration, this may be all one would like to know.

As an example, SPM for the Helium atom with two atoms appears to give a configuration with two half-spherical electron lobes in opposition, as a representative configuration with other equally possible configurations obtained by rotation, as suggested in a previous post and in the sequence Quantum Contradictions.

Instead of seeking information about all possible configurations by solving the 3N dimensional many-electron Schrödinger equation, which is impossible, it may be more productive to seek information about one possible and representative configuration by solving a system of 3 dimensional one-electron wave functions, which is possible.

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