söndag 6 april 2014

Schrödinger's Equation: Smoothed Particle Dynamics

Eigenfunctions of the Hamiltonian for the Hydrogen atom with eigenvalues representing the sum of kinetic and potential energies, with Schrödinger's equation as a smoothed version of the particle dynamics of a harmonic oscillator.  

This is continuation of the previous post How to Make Schrödinger's Equation Physically Meaningful + Computable. Consider the basic case of the Hydrogen atom with one electron (normalized to unit mass and charge):
  • $ih\dot\psi + H\psi =0$,
  • $H\psi =\frac{h^2}{2}\Delta\psi +\frac{1}{\vert x\vert}\psi$,
where $\psi (x,t)$ the complex-valued wave function depending on coordinates of space $x$ and time $t$ with the dot denoting differentiation with respect to time, $H$ is the Hamiltonian operator and $h$ Planck's constant.

In terms of the real part $\phi$ and imaginary part $\chi$ of $\psi =\phi +i\chi$, Schrödinger's equation takes the system form
  1. $h\dot\phi +H\chi =0$,
  2. $h\dot\chi - H\phi =0$.
If $\phi_E(x)$ is an eigenfunction of the Hamiltonian satisfying $H\phi_E =E\phi_E$ with $E$ the corresponding eigenvalue, then the solution can be represented as
  • $\phi (x,t)=\cos(\omega t)\phi_E(x)$,     $\chi (x,t)=\sin(\omega t)\phi_E(x)$, 
with $h\omega =E$, which expresses a periodic exchange between the two real-valued wave functions $\phi$ and $\chi$ mediated by the Hamiltonian $H$.

We can see 1- 2 as an analog of the equation for a harmonic oscillator $\ddot u+\omega^2u=0$ written in system form (with $h=1$)
  • $\dot\phi  + \omega\chi =0$
  • $\dot\chi  - \omega \phi = 0$,
where $\phi =\dot u$ and $\chi =\omega u$, with solution
  • $\phi (x,t)=\cos(\omega t)$,     $\chi (x,t)=\sin(\omega t)$.  
Here the velocity $\phi =\dot u$ connects to kinetic energy $\phi^2 =\dot u^2$ and $\chi =\omega u$ to potential energy $\chi^2 =\omega^2u^2$ and the dynamics of the harmonic oscillation consists of periodic transfer back and forth between kinetic and potential energy with their sum being constant.

Returning now to the Hydrogen atom, we obtain multiplying 1 by $\phi$ and 2 by $\chi$ and integrating in space the following the energy balance
  • $h\frac{d}{2dt}\int\phi^2\, dx + \int \phi H\chi \, dx =0$
  • $h\frac{d}{2dt}\int\chi^2\, dx - \int \chi H\phi\, dx =0$,    
  • $ \int \phi H\chi \, dx = \int \chi H\phi\, dx =\frac{h^2}{2}\int\nabla\phi\cdot\nabla\chi\, dx +\int\frac{\phi\chi}{\vert x\vert}\, dx$,
which shows upon summation (by the symmetry of $H$) that
  • $\frac{d}{2dt}\int\phi^2\, dx =\frac{d}{2dt}\int\chi^2\, dx =0$, 
which allows normalization to  
  • $\int\phi^2\, dx = \int\chi^2\, dx = \frac{1}{2}$,
  • $\int\vert\psi\vert^2\, dx = 1$, for all time. 
Further, multiplying 1 by $\dot\chi$ and 2 by $\dot\phi$ and subtracting the resulting equations shows that
  • $\int (\phi H\phi + \chi H\chi)\, dx$ is constant in time. 
We can now summarize as follows:

A. We see that the solution pair $(\phi ,\chi )$ of 1 - 2 as the real and imaginary part of Schrödinger's wave function $\phi$, represents a periodic exchange mediated by the Hamiltonian $H$ with balancing associated total energies 
  • $\int \phi H\phi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\phi (x,t)\vert^2dx +\int\frac{\phi^2(x,t)}{\vert x\vert}\, dx$,
  • $\int \chi H\chi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\chi (x,t)\vert^2dx +\int\frac{\chi^2(x,t)}{\vert x\vert}\, dx$    
as the sum of kinetic and potential energies.

B. We see that Schrödinger's equation for the Hydrogen atom can be viewed as a smoothed version of a harmonic oscillator with the smoothing effectuated by the Laplacian and with $h$ acting as a smoothing parameter.

C. We see that the system form 1- 2 combines the spatial eigenfunction $\phi_E$ with a periodic time dependence without introducing energy beyond the kinetic and potential energies defined by the Hamiltonian, thus associating these energies to frequency as the essence of quantum mechanics.

D. We see that quantum mechanics and Schrödinger's equation can be given an interpretation which closely connects to classical mechanics, as smoothed particle mechanics, which avoids the common mystifications of particle-wave duality, complementarity, wave function collapse and statistics forced by insistence to use a multidimensional wave function defying a direct physical meaning.

Extension to several electrons can be naturally be made following the idea of smoothed particle dynamics. For details see Many-Minds Quantum Mechanics.

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