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Spherical harmonics color map

  1. Mar 26, 2015 #1

    Maylis

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    Hello,

    I am watching a video about spherical harmonics, and I am at the point where the color map is being shown for various values of ##l## and ##m##

    upload_2015-3-27_2-26-56.png

    My question is, what am I supposed to make of these plots? Pretty colors yes, but what do these things mean?
     
  2. jcsd
  3. Mar 26, 2015 #2

    mathman

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    http://dwb4.unl.edu/Chem/CHEM869Z/CHEM869ZLinks/www.rwc.uc.edu/koehler/biophys/6b.html [Broken]

    It looks like quantum numbers for bound electrons.
     
    Last edited by a moderator: May 7, 2017
  4. Mar 26, 2015 #3
    Spherical harmonics are the equivalent to the Fourier basis on S^2, as both are the eigenfunctions of the natural Laplacian. The Fourier basis is usually labeled with a set of integer wave numbers, k_n, n=1..N where N is the dimension of the coordinate vector space. Spherical harmonics are labeled with two similar integer wave numbers, usually named l and m. The graphs you posted show the function value of the (l,m)th spherical harmonic basis function with brightness as modulus and hue for the complex phase. That should allow you to understand what the complex basis function looks like.
     
  5. Mar 26, 2015 #4
    Not sure what the application is for your example, but the surface of a sphere may have modes of oscillation (radial surface motions).
    Simplest way is for the entire surface to move in and out (the whole sphere gets smaller and bigger) - looks like l=0 m=0
    Another way is for one half to move in as the other half moves out and vice versa - looks like l=1 m=0
    Those are the only ones that look clear in the picture, to me; but the l and m numbers are continuing to identify the modes.
    From there the sphere may be sectioned into four longitudinal quadrants where the opposing pairs make opposing motions.
    Same for eight sections, four in one half and four in the other, etc...
    And others where the moving parts are like the latitudinal bands around the Earth - looks like l=2 m=0

    The sections and bands get complicated because they interact to form whole number counts, some required to be even numbers to support a repeating series of opposite polarities of motion (the longitudinal "slices"), and others not held to that constraint (the bands) because they don't repeat, they occupy the poles... the use of the m and l numbers are used to describe all combination possibilities completely.

    Typically, colors may be used to distinguish which parts are moving in a direction at a moment to mark the polarities of the regions, and boundaries (node lines of no movement) between them.

    These harmonic "motions" may in your example be analogous to some other oscillatory property that may not be a strict motion but rather a change in some other attributes like field strength, density, polarity, probability, etc...
     
  6. Mar 26, 2015 #5
  7. Mar 27, 2015 #6
    Each (l,m) has the function of two variables (angles) with complex value. We can rewrite any complex number in form 'x+iy' to r*Exp(i∆) The colors is due to phase of this complex values (∆) which obtain for any solid angle. This pictures only show phase without amplitudes.
     
  8. Mar 27, 2015 #7
    The images actually do show the magnitude, mapped to the brightness of the color.
     
  9. Mar 27, 2015 #8
    Forgive me. In above picture amplitudes are shown but by the brightness of each color. In first time i don't see it. Therefore this form is appreciate to show spherical functions with complex values.
     
  10. Mar 27, 2015 #9
    Thanks.
     
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