söndag 4 maj 2014

A Three-dimensional Multi-Electron Wave Function

Consider a wavefunction $\psi$  for an atom with $N$ electrons as a sum of $N$ functions $\psi_1(x)$,…,$\psi_N(x)$, all depending on a common three-dimensional space coordinate $x$ (plus time):
  • $\psi (x)=\psi_1(x)+\psi_2(x)+…+\psi_N(x)$,
with associated energy as the sum of kinetic energy, attractive kernel potential energy and repulsive interelectron energy:
  • $E(\psi )= \frac{1}{2}\int\vert\nabla\psi\vert^2dx - \int\frac{N\psi^2}{\vert x\vert}dx+\sum_{j\neq k}\int\int\frac{\psi_j^2(x)\psi_k^2(y)}{2\vert x-y\vert}dxdy$,
under the normalization
  • $\int\psi_j^2dx =1$ for $j=1,...,N$,
where $\psi_j(x)$ represents the distribution of electron $j$.

The ground state is determined as the state of minimal energy determined as the solution of a non-linear system of equations in three space dimensions expressing minimality.  We see that minimization favors atomistic wavefunctions $\psi (x)=\sum_j\psi_j(x)$ built from electronic wave functions $\psi_j$ with disjoint supports, which makes the interelectronic repulsion energy small without cost of kinetic energy.

The ground state of Helium thus will have its two electrons separated into two half-spheres with corresponding wave functions $\psi_1(x)$ and $\psi_2(x)$ meeting smoothly at a common separation surface. It is possible that this is the origin of the Zweideutigkeit or two-valuedness expressed in Pauli's exclusion principle, which Pauli did not like because it was ad hoc without rationale.

The  sequence of posts on Quantum Contradictions explores atomic ground states based on the above wave function with surprisingly good correspondence with observations, see also Many-Minds Quantum Mechanics.

We compare with standard quantum mechanics with multi-dimensional wave functions $\psi (x_1,…,x_N)$ depending on $N$ three-dimensional space coordinates $x_1$,…,$x_N$, typically in the form of a Slater determinant as a linear combination of products of $N$ functions $\psi_1$,…,$\psi_N$, each function separately depending on three space coordinates,  thus based on wavefunctions depending on altogether $3N$ space coordinates. Such multi-dimensional wave functions defy direct physical interpretation and are also impossible to compute for atoms with several electrons and thus do not belong to science. Yet they are supposed to be fundamental to atomistic physics.

The standard view is that macroscopic and microscopic (atomistic) physics are fundamentally different,  because microscopic physics demands a multi-dimensional wave function, while macroscopic physics is described by systems of three-dimensional functions. If also microscopic physics can be described by systems of three-dimensional functions, as indictated above, then there will be no fundamental difference between macroscopic and microscopic physics and a major obstacle for progress can be eliminated.

Computations based on wavefunctions of the above form are under way and will presented when available.  For simple hand calculations see here and here.

PS1 For Helium with two electrons at distance $\frac{1}{2}$ from the kernel and mutual distance $1$ as an approximate ground state configuration energy in the above model, we get $E = -3$, to be compared with the observed $-2.903$.

For Lithium with two electrons at distance $\frac{1}{3}$ from the kernel and mutual distance $\frac{2}{3}$ together with a third electron at distance 1 from an effective kernel of charge +1, we get $E = -8$, to be compared with the observed $-7.5$. The ground state energy of three electrons at distance $\frac{1}{3}$ from the kernel and mutual distance $\frac{1}{2}$, we get $E = -7.5$ indicating that the configuration with two electrons in an inner shell and one in an outer shell has smaller energy and thus is the actual ground state configuration for Lithium, thus obtained without reference to Pauli's exclusion principle.

PS2 Recall that the standard quantum mechanics is formulated in terms of a multi-dimensional wave function $\psi (x_1,x_2,…,x_N)$ depending on $N$ three-dimensional space coordinates $x_1$,…$x_N$, altogheter on $3N$ space coordinates, which is devastating because both physical interpretation and computational determination is impossible. To reduce the dimensionality typically an Ansatz is made as Slater determinants of three-dimensional wave functions $\psi_i$ as linear combinations of products of the form (subject to permutations of the coordinates):
  • $\psi (x_1,…,x_N)=\psi_1(x_1)\psi_2(x_2)….\psi_N(x_N)$,
leading to a set of one-electron wave equations coupled by complex exchange-correlation terms which are very difficult to determine. The above Ansatz with a sum instead of products of three-dimensional wave functions may offer more computationally managable and thus more useful models.

PS3 For Beryllium with 4 electrons, we get $E=-14$ from 2 electrons at distance $\frac{1}{4}$ from the kernel with mutual distance $\frac{1}{2}$, together with $E = -\frac{2}{3}$ from 2 electrons of width $\frac{1}{2}$ at distance $\frac{1}{4} + \frac{1}{2}$ from an effective charge of +2, which gives altogether $E = -14.667 which is exactly what is observed!!

PS4 For N electrons distributed over one shell at distance $\frac{1}{N}$ to the kernel assuming the average distance between any pair of electrons is $\frac{1}{N}$, we get $E = -\frac{N^2}{2}$, which is much larger than the observed $E \approx - N^2$ and thus is not the ground state configuration.  A multi-shell distribution in the model gives better agreement with observations and so the model may capture the real shell structure (without resort to any Pauli exclusion principle).

PS5 Note that the above model allows discontinuous electron distributions (joining smoothy) without cost of kinetic energy which favors electron separation. We compare with Hartree models as systems of one-electron models with continuous electron distributions for which separation requires kinetic energy cost and a resort to Pauli's exclusion principle is necessary to prevent more than two electron distributions to overlay.

PS6 To find the ground state, we can use time-stepping of the parabolic system
  • $\frac{\partial\psi_j(x,t)}{\partial t} = \Delta\psi (x,t) + \frac{N\psi (x,t)}{\vert x\vert}-\sum_{k\neq j}\int\frac{\psi_k^2(y,t)}{2\vert x-y\vert}dy\,\psi_j(t,x)$ for $t > 0$, $j=1,…,N$,
with successive normalization to $\int\psi_j^2(x,t)\, dx=1$ after each time step and $\psi =\sum_{k=1}^N\psi_k$.  Further
  • $V_k\equiv\int\frac{\psi_k^2(y,t)}{2\vert x-y\vert}dy$,
can be computed by solving $-\Delta V_k = 2\pi\psi_k^2$.


4 kommentarer:

  1. What decides the distances that you use in the calculations?

  2. A positive kernel of charge Z will attract an electron of width 1/Z.

  3. Because a one electron ion will have a wave function scaling with exp(-Nr) with the distance r to a positive kernel of charge N.