Sunday, 13 November 2011

spherical harmonics and the 3-sphere

One day in summer of 2007 when I was looking for something interesting to teach those gifted kids at the school I had worked with, a picture (the one below) of the shadow of the 3-sphere grabbed my attention. I gazed upon it and was almost instantaneously reminded of pictures of the spherical harmonics, which I knew from my quantum mechanics courses. I thought, could it be that electrons are circling around the nucleus on a 4 dimensional trajectory. If this resemblance happens to be no coincidence it could make perfect sense that the contradiction between the Bohr model and a spherical cloud of probability suggested by Schrödinger's equation is none at all, since the surface of a 3-sphere is of course a 3 dimensional object. It is needless to say, that there are a lot of open questions (especially for me) yet to be answered. As an example I will give just three (at least for me) open questions:
  1. Why is it, that some picturings of those two resemble one another more than others and what are the underlying rules to that similarities respectively differences?
  2. Is this more or less resemblance still given if we depict them in other geometries i.e. hyperbolic space as necessary to get new insights to quantum-gravity?
  3. What happens to the shortcomings of the Bohr model if it is extended to 4 dimensional  space?
Definitely dispiriting is the fact, that I was not able to find anything on this topic on the internet. I suppose that the probabilistic theory of quantum physics suffices to provide answers to most questions in elementary particle physics such that most people don't bother to endeavour this topic, but nonetheless I find it to be a very attractive one to dig deeper into especially because of the possible insights into quantum-gravity it could provide.
Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line). 
[Source: Wikipedia]

Visual representations of the first few spherical harmonics. For each spherical harmonic, we plot its value on the surface of a sphere (front), and then in polar (back). The polar plot is simply obtained by varying the radius of the previous sphere. [Source: Mayavi]

1 comment:

  1. The electron, as viewed in the Bohr-Model, is after all a consequence of the framework of quantum mechanics. It predicts a kind of standing wave, but not only in space, as Bohr's postulates state, but also in time. It's stationary. Tomorrow, if surrounding parameters are untouched, it'll do the same thing as today.
    This is, what gives rise to the spherical harmonics. They are the mathematical solution to an equation which does not change time (the stationary Schroedinger-equation). This demands symmetry in time.
    In the case of a hydrogen atom, the only case where spherical harmonics are actually a solution, the equation can be symmetrized even more. It's completly symmetric in all spatial dimensions.
    So it seems like we need a solution which is symmetrical not only in time or space alone, but in both, and thats where the spherical harmonics comes in. They are the mathematical objects with the desired properties: 4-Dimensional Symmetry (don't know if there a other, would be an interesting question).
    Time, well, is another thing in quantum mechanics... :)