Density of classical states in quantum theory

Question

Let's first treat electrons as classical objects. I can evaluate the classical energy of each state in a configurational space (3N real numbers and, say, spins) using just Coulomb's law.

Then I calculate the electronic wavefunction of the ground state, $\phi_0(\mathbf{r})$, in each point of the configurational space (classical state), which will give me the (square of amplitude...) probability density of each such classical state.

Combining these two results, I get the "density of classical states" of the ground state electronic wavefunction - the probability as a function of the classical energy, $c(E)$. I think this $c(E)$ might have some general features which might be used for speeding up post-Hartree-Fock (HF) quantum chemistry calculations.

I am not a theoretical physicists, therefore I do not know, how to ask Google / WoK, etc. Is there any study about this topic? Is there such an approach, and if so, what are its merits?

Answer 1

I think you should define your problem a little more precise. I now understand you have N particles of the same charge, probably confined to some region (otherwise they will dissipate to infinity). If there is some neutralizing background, then you are lead to "Wigner crystals".

Now - for the quantum problem - I assume you still want to treat N interacting quantum particles, in the same physical setup (some boundaries and -- if needed -- neutralizing background). This is generally a very difficult problem, and the HF-method you mention is one of the approaches.

Anyways -- your "wavefunction" is than a function of "3N" variables (3 for each constituent), and the "amplitude" of it is not very telling. You might project from it, to get "probability of finding a particle" at each place (x,y,z), and this is what most would consider to be probability density of the fluid (formed by the particles).

The above is for the ground state, that is the state of lowest energy, that is the lowest eigenvalue of the Hamiltonian. In general the system will have a complicated spectrum of excitations (think of simple atoms, like Helium, or Lithium, where the spectrum is already very rich, although you only have a few electrons). Due to the confinement, the spectrum will likely be discrete, i.e. will consist of (irregularly spaced) points on the real line. You might form a histogram of this set, and obtain an averaged ``distribution of energies = eigenvalues of Hamiltonian''; for one particle in a box with periodic "holes" you obtain a form of "crystalline band structure" in this way. For your fluid -- it will be rather difficult to obtain this information (I do not know how to compute such a spectrum in general, i.e. without assuming, that the system is rarefied, and that the interactions are weak).

But in general -- the quantum problem will not have much in common with energies of classical distributions of particles, I believe.