| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
04 Nov 2006 11:25:19 AM |
| Object: |
lie groups |
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?
thanks jf
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| User: "Timo A. Nieminen" |
|
| Title: Re: lie groups |
04 Nov 2006 03:45:02 PM |
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On Sat, 4 Nov 2006, 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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| User: "" |
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| Title: Re: lie groups |
04 Nov 2006 07:46:44 PM |
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The group of rotatations of 3D space is SO(3). When you rotate a
physical system, this rotation corresponds to an element of SO(3). The
wavefunction of an electron will obviously change, and this constitutes
an `action' of SO(3) on the space of electron wavefunctions.
It is a mathematical fact that `representations' of SO(3) can only be
odd-dimensional. That means that given any electron wavefunction, it
will belong to a set of 1, 3, 5, 7,... independent wavefunctions that
can all be rotated into each other.
From a more physical point of view (although it all comes out of the
above mathematics), you can look at it as follows. The orbital angular
momentum of an electron is quantised; it can only assume the values
h-bar*sqrt(j(j+1)), where j = 0, 1, 2, 3,... The value of a single
chosen component of the angular momentum is also quantised; it can take
the values m*h-bar, where m = -j, -j + 1, ..., j - 1, j. This is
always an odd number. Because rotating a system can't change the total
angular momentum, it only changes m. This is why when a system is
rotated, the wavefunctions rotate amongst themselves in sets of 1, 3,
5, 7, ...
I hope that helps!
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| User: "" |
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| Title: Re: lie groups |
08 Nov 2006 09:15:37 PM |
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It does,thanks. jk
On 4 Nov 2006 17:46:44 -0800, wrote:
The group of rotatations of 3D space is SO(3). When you rotate a
physical system, this rotation corresponds to an element of SO(3). The
wavefunction of an electron will obviously change, and this constitutes
an `action' of SO(3) on the space of electron wavefunctions.
It is a mathematical fact that `representations' of SO(3) can only be
odd-dimensional. That means that given any electron wavefunction, it
will belong to a set of 1, 3, 5, 7,... independent wavefunctions that
can all be rotated into each other.
From a more physical point of view (although it all comes out of the
above mathematics), you can look at it as follows. The orbital angular
momentum of an electron is quantised; it can only assume the values
h-bar*sqrt(j(j+1)), where j = 0, 1, 2, 3,... The value of a single
chosen component of the angular momentum is also quantised; it can take
the values m*h-bar, where m = -j, -j + 1, ..., j - 1, j. This is
always an odd number. Because rotating a system can't change the total
angular momentum, it only changes m. This is why when a system is
rotated, the wavefunctions rotate amongst themselves in sets of 1, 3,
5, 7, ...
I hope that helps!
.
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