From: "Michael Weiss" Subject: Re: Matrix mechanics Date: Fri, 16 Jun 2000 08:42:03 -0400 Newsgroups: sci.physics.research Summary: [missing] Gerard Westendorp wrote: | But matrix mechanics was there before Schrodinger. | So how did Heisenberg's original Matrix mechanics | work? The Dover book _Sources of Quantum Mechanics_ has an excellent historical introduction by its editor, van der Waerden, that focusses on this question. (Plus of course the original papers, translated into English.) And being a Dover book, it's cheap. David Cassidy's Heisenberg bio, _Uncertainty_, also sheds light on this question. Here's the short answer. (How many times have I seen good reading recommendations on the net, and ignored them?) Classical dispersion theory treated the interaction of an atom with the EM field. It made heavy use of Fourier expansions, e.g., you might assume the atom contained an oscillating charged particle with position variable x: x = sum a_m exp (imwt) and you'd have all these formulas involving x. For example, the energy is proportional to x^2, you have a differential equation for x, you have formulas relating the damping term to the radiation emitted, etc. These formulas turn into relations among various Fourier coefficients. For example, the energy involves x^2, say: x^2 = sum b_m exp(imwt) and you can express the b_m in terms of the a_m; the Coulomb force involves a more complicated term, whose Fourier coefficients nonetheless are related to the Fourier coefficients of x; etc. Since classical dispersion theory got some things right but clearly had problems, Heisenberg (and Born and Kramers) were looking for a way to quantize it. (They thought and wrote in these terms at the time.) Heisenberg said: the variable x is not observable, only the spectra are observable. Let's consider the spectral line for a transition from level m to level n --- let's assume we have a coefficient a_{mn} which describes this. In other words, let's try to redo classical dispersion theory, replacing the a_m with the a_{mn}. So the matrix a_{mn} is our quantum version of the variable x. Then he started playing around with the classical equations. He began with a toy example, the anharmonic oscillator given by the equation d^2x/dt^2 + w^2 dx/dt + cx^2 = 0, and guided by the classical model came up with the formula for matrix multiplication. I.e., letting b_{mn} represent x^2, he cooked up a formula for the b_{mn} in terms of the a_{mn}. You'll notice that his anharmonic equation has a term cx^2 in it. I say he "came up with" the matrix multiplication formula. He couldn't really *derive* it, of course. But he did have an important clue: the Ritz combination principle in spectroscopy said that if v_{ij} is the frequency of the transition from lines i to j, and likewise for v_{jk}, then: v_{ik} = v_{ij} + v_{jk} The non-commutativity was obtained by working backwards, so to speak. Heisenberg noticed that his multiplication was non-commutative. Born assured him that was OK, and soon guessed the formula for pq-qp. Pascual Jordan then derived this equation (from the equations of motion, plus some reasonable assumptions), and Dirac finally recast the whole theory more abstractly. We would say that Dirac realized the real subject matter was linear operators in a Hilbert space, and the matrices were just a representation, but Dirac, like many of the really great physicists, preferred to reinvent math rather than learning it elsewhere... ============================================================================== From: gdpusch@NO.xnet.SPAM.com (Gordon D. Pusch) Subject: Re: Gaussian pulse in 2D FDTD mesh Date: 10 Aug 2000 17:26:45 -0500 Newsgroups: sci.physics.electromag,sci.math.num-analysis Summary: [missing] Russell Iain Macpherson writes: > Jos Bergervoet wrote: >> >> But there is also dispersion in the wave equation itself for d=2, which >> you should see even if your mesh size goes to zero (to make the numerical >> dispersion vanish). Only for d=1 and for d=3 will the wave equation >> preserve signal shape. [...] > Thanks very much for the reply. It has all made me take a much closer > look at this area. I did not quite understand what you meant by only 1D > and 3D situations will the wave shape be maintained. [...] > Perhaps I am misunderstanding something here? Dispersion should happen > in both 2D and 3D cases, but can be avoided totally in the 1D case. And > to minimise dispersion in the 2D and 3D cases the cell size and time > step size should be minimised. Jos is referring to an old result by (IIRC) Sommerfeld that non-dispersive wave-propagation only occurs in spaces with an _ODD_ number of dimensions, due to the properties of the Green's function of the wave equation. In two dimensions, for example, the Green function for the wave equation has a Bessel-function-like ``tail'' trailing behind the wavefront proper. IIRC, the argument was quite elegant because it noted that the Green function for the wave equation in an EVEN number of spatial dimensions can always be related to the solution of the wave equation for an infinitely long impulsive ``line source'' in a space of ONE MORE DIMENSION; hence, the Green function in any space with even dimensionality must have an infinitely long ``tail'' that asymptotically decays with a ~1/(t - t_0)^N power-law envelope... -- Gordon D. Pusch perl -e '$_ = "gdpusch\@NO.xnet.SPAM.com\n"; s/NO\.//; s/SPAM\.//; print;'