Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What is the simplest (lowest order) real spherical harmonic that cannot be rotated into its own mirror image? I would like to make an illustration of a simple smooth chiral object, but cannot find a good example.

Edit: I should have clarified that I need a linear combination of spherical harmonics, not just one of them.

share|improve this question
1  
I'm not sure if I understand chirality correctly, but maybe you could take a linear combination of basic harmonics composed with incommensurate rotations? –  anon Aug 26 '11 at 9:58
    
Apologies for my incorrect answer; you're quite right that linear combinations of $l=1$ harmonics can be rotated into their mirror images. Adding an $l=0$ function won't help, so that shows you'll need at least $l=2$. An arbitrary linear combination of the nine functions up to $l=2$ is chiral (even three $l=1$ and one $l=2$ should suffice), so that raises the question how exactly you define "simplest", since "lowest order" doesn't distinguish between the various possibilities including $l=2$ functions. –  joriki Aug 26 '11 at 14:10
    
Thanks for your comment. I don't think arbirary combinations are chiral, for exampel you can make the functions independent of z and then they are not chiral. But maybe most functions are indeed chiral. For example we can take the harmonic polynomial $x+y+z + xy$, can you prove that this is chiral? I am also interested in how to prove these things. –  uekstrom Aug 26 '11 at 14:41
    
I've updated and undeleted my answer. About "arbitrary combinations": By "arbitrary" I meant something like "in general position". As it happens, your example is in the set of measure zero of achiral combinations :-) –  joriki Aug 29 '11 at 9:20
add comment

1 Answer

Again, apologies for the initial incorrect answer. I hope I got it right this time. :-)

Here's a way to determine rigorously whether a given linear combination of spherical harmonics is chiral. The functions with different values of $l$ don't get mixed under rotations. Thus, a combination is achiral if and only if there's a single rotation that rotates the mirror images of all $l$ components into their originals. So you can look at the $l$ values individually and see whether there's any intersection among the sets of rotations.

For your example, $x+y+z+xy$, the $l=1$ component $x+y+z$ is a rotated version of $Y_{10}$ oriented towards $(1,1,1)$. The rotations that rotate the mirror image of this into the original are rotations through $\pi$ about an axis perpendicular to $(1,1,1)$. (Subsequently rotating about $(1,1,1)$ doesn't add anything; it just corresponds to a different choice of axis for the first rotation.)

The $l=2$ component $xy$ is invariant under inversion, so we need to find the rotations that leave it invariant. Its angular dependence is $\sin^2\theta\sin\phi\cos\phi$, and the only rotations that leave the $\theta$ dependence invariant are ones with axes in the $x$-$y$ plane or orthogonal to it (since the poles have to end up at the poles). There's one axis in the $x$-$y$ plane that's orthogonal to $(1,1,1)$, namely $(1,-1,0)$. Coincidentally, rotating through $\pi$ about that axis corresponds to $x\to-y$, $y\to-x$, $z\to-z$, and thus works for $x+y+z+xy$. However, this is only one of two axes in the $x$-$y$ plane for which rotating through $\pi$ leaves $xy$ invariant, since rotating the rotation axis about the $z$ axis by $\alpha$ amounts to rotating the result about the $z$ axis by $2\alpha$, and that only leaves $xy$ invariant if $2\alpha$ is a multiple of $\pi$. So you can pick almost any other combination of the $l=1$ functions, as long as the associated vector isn't orthogonal to either the $z$ axis or the axes $(1,-1,0)$ and $(1,1,0)$, and add $xy$ to get a chiral combination.

share|improve this answer
    
Yes, sorry. I want a combination of spherical harmonics with different L values so that the total function is chiral. –  uekstrom Aug 26 '11 at 11:10
    
You can always rotate a sum of first order harmonics into its negative, it's just to take a 180 degree rotation in a plane containing (a,b,c). So that won't cut it. –  uekstrom Aug 26 '11 at 11:33
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.