DATA-log

You can read all sort of new age weirdness and academic pomposity associated with quantum mechanics. Controversy rages on up to this day about the fundamentals of this cornerstone of modern physics.

So what is so special about QM. My interest in it comes from the mathematical framework it was built on. I think it’s an interesting way to learn new mathematics, and indeed it has proven to be just that. My experience has been that once you finally get the hang of it, all the mystery you’ve read about it seems to diffuse in to thin air.

It is easy to deflate QM, if for some strange reason that is your goal. The basic premise the quantum is built on is diffusion. Yes, the Schrödinger wave equation is nothing more than a mathematical description of diffusion in imaginary time. All the nonintuitive aspects of QM are closely related to the complex plane. Schrödinger equation was an important step in the development of the theory, but is actually just a small part of the framework and it is incomplete without the concept of what a measurement is.

Of all the books written about QM I’ve studied, the best introduction by far is in the good old communist era Landau and Lifshitz book Course of Theoretical Physics Volume 3: Quantum Mechanics (Non-relativistic Theory). It gets right to the point; it is impossible to measure properties of fundamental particles with absolute precision. The reason is simple and intuitive. Measurements of fundamental particles have to be made with some sort of force mediating way. In the case of measuring the position of an electron you can use light. The more precision you want, the shorter wavelength you’ll end up using. This in turn means that the photon carries more energy. When the photons interact and scatter with the electron you want to measure, the more the electron recoils from this energy. Repeated measurements will end up jittering the particle more and more, adding to the uncertainty of the measured path. Consecutive lower energy, bigger wavelength photons end up jittering the particle’s path less, but the wavelength dictates the uncertainty in the measurement.

This ends up being reproduced by the basic QM vector space framework. The Heisenberg uncertainty principle in the mathematics stems from the commutation relations of the observables, which in the simplest case here are the position and momentum of the particle.

The above video illustrates the unitary time evolution and diffusion of a simple potential well quantum system in both the position and momentum spaces. The initial gaussian probability distribution moves and spreads out in space with time. The distribution cancels out with itself in this bound system, in free space the wave packet would diffuse indefinetly. You don’t actually exactly know where the particle is until you measure it, and collapse the wave function to some specific state. For example measuring the position would pick a definite position from the distribution at random with some probability and the time evolution would diffuse from that point on, until a new measument is made.

Is it then such a mystery that QM manages to be backed up by experimental measurements, when the whole theory is rigged up to do exactly that?

Quantum sells in the supermarket of ideas.

Studying wave equation discretization has personally led me to understand and appreciate the physics which rely heavily on this mathematics, namely electrodynamics and field theories. I think this sort of ‘visual intuition’ is a important part of learning when dealing with dynamics which can be extremely complicated and complex, at least it has been exactly that  for me. Studying the computability of these equations has definetly paved some way in these theoretically dense subjects.

I recently made few basic computer models of the Schrödinger wave equation with MATLAB for computational physics course work.

Here is a two dimensional version of the discretisized time-dependent wave equation calculated using a clever leapfrog integration algorithm scheme by Visscher.

Things start to get visually more interesting when integrating the time-dependant equation in three dimensions.

Here is another run with a slightly different value for the momentum of the wavepacket.

Unfortunately MATLAB seriously lacks in the volumetric plotting department so we’ll have to do with phong shaded isosurfaces instead of a more appropriate voxel based plot. Don’t worry, this will not end up forming black holes and destroying Vulcan.

V. S. Zuev, G. Ya. Zueva
(Submitted on 3 Nov 2008 (v1), last revised 4 Nov 2008 (this version, v2))

Abstract: The paper is of a methodological character and has as a goal to give a brief description of the concept of theory and practical application of very slow optical plasmons. They exist on the metal-dielectric boundaries, namely, on very thin metal films and fibers and as standing waves on metal spheres and ellipsoids. The material presented in the paper features by widening the common concepts of electromagnetic modes of various spaces, of the probability of spontaneous emission, of creation of optical images, of the limits of optical focusing, and of the photon linear momentum. All mentioned studies are completed in recent years. The problem of the photon momentum in a dielectric medium was the topic of irreconcilable disputes for 100 years starting in the time of appearing of Minkowski and Abraham famous papers. Various practical applications are surveyed: the experiments with a great intensification of an atom spontaneous emission into a plasmonic field mode of a metal nanoparticle, the experiments on focusing optical radiation into a spot that substantially smaller than a diffraction limited spot, a so called near perfect Pendry lens that produces the image with details that substantially smaller than defined by diffraction, and lastly, the concept of hundredfold and more magnification of a photon mechanical linear momentum in a plasmon. The work completed is supported by RFBR, grants Nos 05-02-19647, 07-02-01328.

http://arxiv.org/abs/0811.0810

Craig F. Bohren
Department of Meteorology, Pennsylvania State University, University Park, Pennsylvania 16802

(Received 11 February 1982; accepted 26 April 1982)

A particle can indeed absorb more than the light incident on it. Metallic particles at ultraviolet frequencies are one class of such particles and insulating particles at infrared frequencies are another. In the former strong absorption is associated with excitation of surface plasmons; in the latter it is associated with excitation of surface phonons. In both instances the target area a particle presents to incident light can be much greater than its geometrical cross-sectional area. This is strikingly evident from the field lines of the Poynting vector in the vicinity of a small sphere illuminated by a plane wave. ©1983 American Association of Physics Teachers

Link to citation.

Having studied the fundamentals of quantum mechanics from the perspective of numerical computation it has become obvious that stability is a major drawback in most fast numerical approaches. The best scheme to tackle with the issue is by Visscher whose explicit algorithm is both fast and stable. Accuracy is not my biggest concern since I’m mostly interested in visualization of the quantum mechanical world but the algorithm holds very well. The algorithm is second order accurate, there is some dispersion of the wave packet however. It is also straightforward to extend it to n-dimensional space.

My processing example applet of the 1D time-dependent Schrödinger wave equation with the source code is at http://www.punainen.org/~biotek/visscher1d

Reference to the original paper: P.B. Visscher “A fast explicit algorithm for the time-dependent Schrödinger equation”, Computers in Physics Nov/Dec 1991, 596-8