Re: lie groups

On Sat, 4 Nov 2006, jfisher@xxxxxxxxxxxxxxxxxx wrote:

I'm currently reading a book entitled "Symmetry Monster " and have
reached a section were he is discussing the connection between lie
algebra and electron orbitals. Now from what I understand of lie
theory,
the generators come from the parameters of a transformation. So for
example, in three dimensions ,a symmetry transformation involving 4
parameters e.g. a,b,c,d would have four generators for the algebra. I
need to quote so I can explain my confusion. " But most electron
orbits in an atom are not spherically symmetric, and the group of
rotations can change one into another. In this case the operation of
the
group is more than one -dimensional -there is more than one degree of
freedom. The number of degrees of freedom -or mathematically speaking
the number of dimensions - has to be an odd number 1,3,5,7 ect.. This
is
mathematical fact about a lie group of rotations in three dimensions.
"

So what are we dealing with here? My first guess would be that
there
exists a class of symmetry transformations of Schrodingers equation
that require either 1,3,5 or 7 parameters for the transformation to
exist and that no such symmetry transformations exist for
transformations that contain an even number of parameters. How far off
am I here? If I'm way off , what exactly is he talking about here?

The 1,3,5,7, ... turns up in most spherical coordinate PDE problems where the radial and angular parts are separable. The angular part typically turns out to be spherical harmonics, Y_nm. Generally, the degree n is related to the radial behaviour, and the order m is sort-of independent of the radial behaviour. Thus, rotations can't change n, but will change m. Since -n <= m <= n, you get the 1,3,5,7 ... . For equations with this type of symmetry, the 1,3,5,7 etc functions R(r)*Y_nm(theta,phi) of varying m for a given n are a complete basis set of solutions with that n. How this relates to Lie groups I don't know.

There's a nice (but brief) discussion about the symmetry of these in Landau & Lifshitz, Classical theory of fields, in the context of the relationship between symmetry of Cartesian multipoles and spherical multipoles (ie in Cartesian coordinates, you have 1 monopole, 3 dipoles, 9 quadrupoles, 27 octupoles etc at first glance, but these have to be equivalent to the 1,3,5,7 etc spherical multipoles).

--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html

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