# Quantum Mechanics as Smoothed Particle Mechanics

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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
• $K(\Psi )=\frac{1}{2}\sum_{j=1}^N\int\vert\nabla_j\Psi\vert ^2dx\, ,$

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

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
• $K=\frac{1}{2}\sum_{j=1}^N\int\vert\nabla_j\Psi_j\vert ^2dx_j$ .

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 $\beta =1$).

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