Quantum Mechanics as Smoothed Particle Mechanics

· Uncategorized
Authors

Schrödinger’s equation as the basic mathematical model of quantum mechanics of a system of N interacting particles (electrons and kernels), is commonly motivated as an analog to a classical N-body problem with classical momentum translated to the spatial gradient of a 3N-dimensional wave function

  • \Psi (x) = \Psi (x_1, x_2,..., x_N)

depending on N 3-dimensional space variables x_1,..., x_N,  with the dependence on x_j supposed to describe the whereabouts of particle j. The kinetic energy of particle j is then represented by

  • \frac{1}{2}\vert\nabla_j\Psi\vert ^2 

with \nabla_j the gradient with respect to x_j. This is viewed as an ad hoc model introducing a new form of kinetic energy and \vert\Psi (x) \vert ^2 interpreted as the probability distribution of the particle configuration with particle j located at x_j,\, j = 1,..., N.

The ground state of an atom formed by N negative electrons surrounding a positive kernel of charge N fixed at the origin, is given by the wave function \Psi (x) minimizing the total energy E(\Psi ) = P(\Psi )+ K(\Psi ) as the sum of potential energy P(\Psi ) and kinetic energy K(\Psi ) with dx = dx_1...dx_N

  • P(\Psi )= -\sum_{j=1}^N\int V_j\vert\Psi\vert^2\, dx + \sum_{i<j}\int V_{ij}\vert\Psi\vert^2dx\, ,
  • K(\Psi )=\frac{1}{2}\sum_{j=1}^N\int\vert\nabla_j\Psi\vert ^2dx\, ,

where

  • V_j=\frac{N}{\vert x_j\vert}\, ,
  • V_{ij}=\frac{1}{\vert x_i - x_j\vert}\, .

We compare with the potential energy P_\delta (x) =P_\delta (x_1,...,x_n) of N unit point charges located at the points x_j,\, j=1,..., N:

  • P_\delta (x) =-\sum_{j=1}^N\frac{N}{\vert x_j\vert} + \sum_{i<j}\frac{1}{\vert x_i -x_j\vert}

formally obtained by replacing \vert\Psi\vert^2 by a sum of delta functions at the points x_j.

In this perspective the potential energy P is obtained from P_\delta by gradient regularization of the delta functions with the regularization named kinetic energy.

We are thus led to view ground state quantum mechanics as a regularized or smoothed version of deterministic classical particle mechanics, and it is then natural to view the multidimensional wave distribution \vert \Psi (x)\vert^2 as the sum of regularized single electron 3-dimensional wave distributions \vert\Psi_j(x_j)\vert^2 with

  • P = -\sum_{j=1}^N\int V_j\vert\Psi_j\vert^2\, dx_j + \sum_{i<j}\int V_{ij}\vert\Psi_i\vert^2\vert\Psi_j\vert^2dx_idx_j
  • K=\frac{1}{2}\sum_{j=1}^N\int\vert\nabla_j\Psi_j\vert ^2dx_j .

This corresponds to a Hartree model as a system of single electron wave functions \Psi_j(x_j) with ground state defined by minimization of the total energy P + K under the normalization

  • \int\vert\Psi_j(x_j)\vert^2dx_j =1 for j=1,..., N.                         (1)

Note that to refer to K as kinetic energy is confusing since it suggests motion where no motion is involved. It is more natural to view K as a form of elastic energy related to the amount of “compression” of the electron as expressed by the regularization term.

Viewing quantum mechanics as smoothed particle mechanics allows both computational solution and deterministic physical interpretation thus avoiding the severe difficulties of playing with a multi-dimensional wave function which is uncomputable and does not describe physical reality. It also opens to modeling with different forms of regularization since regularization is not cut in stone in the same way as the standard multi-dimensional wave function which does not allow any modification since it is chosen ad hoc and does not connect to physical reality.

The ground state can be computed by an iterative method defined by explicit time-stepping of the following system of single electron problems: Find \Psi_j(x_j) for j=1,...,N satisfying for j=1,...,N, and all x_j:

  • \frac{\partial \Psi_j}{\partial t} = V_j\Psi_j -W_j\Psi_j + \frac{1}{2}\Delta_j\Psi_j,
  • W_j(x_j) =\frac{1}{2}\sum_{i\neq j}\int V_{ij} \vert\Psi_i\vert^2dx_i,

where in each time step the normalization condition (1) is restored and \Delta_j is the Laplacian with respect to x_j and t is a fictitious time variable. The electronic repulsion potential W_j can be computed by solving the Poisson equation:

  • \Delta_jW_j =2\pi \sum_{i\neq j}\int V_{ij} \vert\Psi_i\vert^2dx_i .

The challenge is now to explain the periodic table with its electron shell structure from the above single electron system, without use of Pauli’s exclusion principle, which is an ad hoc assumption without physical justification. Experience with Helium then suggests that for electrons i and j in the same atomic shell, \Delta_i\Psi_i and \Delta_j\Psi_j should better be replaced by their mean value , in the time stepping towards the minimum energy of the ground state reflecting angular averaging within each shell.

For Helium the time stepping could start from the decentered electron configuration described here (with \alpha = 2 and \beta =1).

1 Comment

Comments RSS
  1. NWA

    I guess the next challenge is not to show the periodic table, but, if your theory allows for stable bulk matter.

    If it doesn’t, just scrap it because it is then useless.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: