New Quantum Mechanics 17: The Nightmare of Multi-Dimensional Schrödinger Equation

Once Schrödinger had formulated his equation for the Hydrogen atom with one electron and with great satisfaction observed an amazing correspondence to experimental data, he faced the problem of generalising his equation to atoms with many electrons.

The basic problem was the generalisation of the Laplacian to the case of many electrons and here Schrödinger took the easy route (in the third out of Four Lectures on Wave Mechanics delivered at the Royal Institution in 1928) of a formal generalisation introducing a set of new independent space coordinates and associated Laplacian for each new electron, thus ending up with a wave function $\psi (x1,...,xN)$ for an atom with $N$ electrons depending on $N$ 3d spatial coordinates $x1$,...,$xN$.

Already Helium with a Schrödinger equation in 6 spatial dimensions then posed a severe computational problem, which Schrödinger did not attempt to solve.  With a resolution of $10^2$ for each coordinate an atom with $N$ electrons then gives a discrete problem with $10^{6N}$ unknowns, which already for Neon with $N=10$ is bigger that the total number of atoms in the universe.

The easy generalisation thus came with the severe side-effect of giving a computationally hopeless problem, and thus from scientific point meaningless model.

To handle the absurdity of the $3N$ dimensions rescue steps were then taken by Hartree and Fock to reduce the dimensionality by restricting wave functions to be linear combinations of products of one-electron wave functions $\psi_j(xj)$ with global support:

• $\psi_1(x1)\times\psi_2(x2)\times ....\times\psi_N(xN)$    

to be solved computationally by iterating over the one-electron wave functions. The dimensionality was further reduced by ad hoc postulating that only fully symmetric or anti-symmetric wave functions (in the variables $(x1,...,xN)$) would describe physics adding ad hoc a Pauli Exclusion Principle on the way to help the case. But the dimensionality was still large and to get results in correspondence with observations required ad hoc trial and error choice of one-electron wave functions in Hartree-Fock computations setting the standard.

We thus see an easy generalisation into many dimensions followed by a very troublesome rescue operation stepping back from the many dimensions. It would seem more rational to not give in to the temptation of easy generalisation, and in this sequence of posts we explore such a route.

PS In the second of the Four Lectures Schrödinger argues against an atom model in terms of charge density by comparing with the known Maxwell's equations for electromagnetics in terms of electromagnetic fields, which works so amazingly well, with the prospect of a model in terms of energies, which is not known to work.