torsdag 28 maj 2015

Physical Quantum Mechanics: Time Dependent Schrödinger Equation

We consider a Schrödinger equation for an atom with $N$ electrons of the normalized form: Find a wave function
  • $\psi (x,t) = \sum_{j=1}^N\psi_j(x,t)$
as a sum of $N$ electronic complex-valued wave functions $\psi_j(x,t)$, depending on a common 3d space coordinate $x$ and time coordinate $t$ with non-overlapping spatial supports $\Omega_1(t)$,...,$\Omega_N(t)$, filling 3d space, satisfying
  • $i\dot\psi (x,t) + H\psi (x,t) = 0$ for all $(x,t)$,       (1)
where the (normalised) Hamiltonian $H$ is given by
  • $H(x) = -\frac{1}{2}\Delta - \frac{N}{\vert x\vert}+\sum_{k\neq j}\int\frac{\vert\psi_k(y,t)\vert^2}{2\vert x-y\vert}dy$  for $x\in\Omega_j(t)$,
and the electronic wave functions are normalised to unit charge:
  • $\int_{\Omega_j}\vert\psi_j(x,t)\vert^2 =1$ for all $t$ for $j=1,..,N$.
The total wave function $\psi (x,t)$ is thus assumed to be continuously differentiable and the electronic potential of the Hamiltonian acting in $\Omega_j(t)$ is given as the attractive kernel potential together with the repulsive kernel potential resulting from the combined electronic charge distributions $\vert\psi_k\vert^2$ for $k\neq j$.

The Schrödinger equation in the form (1) is a free-boundary problem where the supports $\Omega_j(t)$ of the electronic wave functions may change over time.

We solve (1) by time-stepping the system
  • $\dot u + Hv = 0$, $\dot v - Hu = 0$       (2)
obtained by splitting the complex-valued wave function $\psi = u+iv$ into real-valued real and imaginary parts $u$ and $v$ (and with $\vert\psi\vert^2 =u^2+v^2$.) 

This is a free-boundary electron (or charge) density formulation keeping the individuality of the electrons, which can be viewed as a "smoothed $N$-particle problem" of interacting non-overlapping "electron clouds" under Laplacian smoothing. The model (1) connects to the study in Quantum Contradictions showing a surprisingly good agreement with observations.

In particular, the time-dependent form (2) is now readily computable as a system of wave functions depending on a common 3d space variable and time, to be compared to the standard wave equation in $3N$ space dimensions which is uncomputable.

I am now testing this model for the atoms in the second row of the periodic table, from Helium (N=2) to Neon (N=10), and the results are encouraging: It seems that time dependent N-electron quantum mechanics indeed is computable in this formulation and the model appears to be in reasonable agreement with observations.  This gives promise to exploration of atoms interacting with external fields, which has been hindered by uncomputability with standard multi-d wave functions.

PS The formulation readily extends to electrodynamics with the Laplacian term of the Hamiltonian replaced by

  •  $\frac{1}{2}(i\nabla + A)^2$
and the potential augmented by $\phi$, where $A=A(x,t)$ is a vector potential, $\phi =\phi (x,t)$ is a scalar potential with $E = -\nabla\phi -\dot A$ and $B=\nabla\times A$ given electric and magnetic fields $E=E(x,t)$ and $B=B(x,t)$ depending on space and time. 

Inga kommentarer:

Skicka en kommentar