onsdag 1 juni 2016

New Quantum Mechanics 1 as Classical Free Boundary Problem

Let me (as a continuation of the sequence of posts on Finite Element Quantum Mechanics 1-5) present an alternative formulation of the eigenvalue problem for Schrödinger's equation for an atom with $N$ electrons starting from an Ansatz for the wave function
• $\psi (x) = \sum_{j=1}^N\psi_j(x)$      (1)
as a sum of $N$ electronic real-valued wave functions $\psi_j(x)$, depending on a common 3d space coordinate $x\in R^3$ with non-overlapping spatial supports $\Omega_1$,...,$\Omega_N$, filling 3d space, satisfying
• $H\psi = E\psi$ in $R^3$,       (2)
where $E$ is an eigenvalue of the (normalised) Hamiltonian $H$ given by
• $H(x) = -\frac{1}{2}\Delta - \frac{N}{\vert x\vert}+\sum_{k\neq j}V_k(x)$ for $x\in\Omega_j$,
where $V_k(x)$ is the potential corresponding to electron $k$ defined by
• $V_k(x)=\int\frac{\psi_k^2(y)}{2\vert x-y\vert}dy$, for $x\in R^3$,
and the wave functions are normalised to correspond to unit charge of each electron:
• $\int_{\Omega_j}\psi_j^2(x) dx=1$ for $j=1,..,N$.
One can view (2) as a formulation of the eigenvalue problem for Schrödinger's equation, starting from an Ansatz for the total wave function as a sum of electronic wave function according to (1), as a classical free boundary problem in $R^3$, where the electron configuration is represented by a partition of $R^3$ into non-overlapping domains representing the supports of the electronic wave functions $\psi_j$ and the total wave function $\psi$ is continuously differentiable.

Defining $\rho_j = \psi_j^2$, we have
• $\psi\Delta\psi = \frac{1}{2}\Delta\rho-\frac{1}{4\rho}\vert\nabla\rho\vert^2$,
and thus (2) upon multiplication by $\psi$ takes the form
• $-\frac{1}{4}\Delta\rho+\frac{1}{8\rho}\vert\nabla\rho\vert^2-\frac{N\rho}{\vert x\vert}+V\rho = E\rho$ in $R^3$,                   (3)
where
• $\rho_j\ge 0$, $support(\rho_j)=\Omega_j$ and $\rho_j=0$ else,
• $\int_{\Omega_j}\rho_jdx =1$,
• $\rho =\sum_j\rho_j$,
• $V\rho=\sum_{k\neq j}V_k\rho_j$ in $\Omega_j$,
• $\Delta V_j=2\pi\rho_j$  in $R^3$.
The model (3) (or equivalently (2)) is computable as a system in 3d and will be tested against observations. In particular the ground state of smallest eigenvalue/energy E is computable by parabolic relaxation of (3) in $\rho$. Continuity of $\psi$ then corresponds to continuity of $\rho$.

We can view the formulation (3) in the same way as that explored for gravitation, with the potential $V_j$ primordial and the electronic density $\rho_j$ defined by $\rho_j =\frac{1}{2\pi}\Delta V_j$ as a derived quantity, with in particular total electron-electron repulsion energy given by the neat formula
• $\frac{1}{2\pi}\sum_{k\neq j}\int V_k\Delta V_jdx=-\frac{1}{2\pi}\sum_{k\neq j}\int\nabla V_k\cdot\nabla V_jdx$
in terms of potentials with an analogous expression for the kernel-electron attraction energy.

For the choice of free boundary condition see later post.