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Where to find spherical harmonics on a three-sphere?
Date: 1997/11/09
The spherical harmonics Y_lm(theta,phi) show up a lot in quantum
mechanics, for example the eigenfunctions for a particle constrained to
move on the surface of a two-sphere. I am curious what the
eigenfunctions look like for a particle which lives in four dimensions
but is constrained to move on the surface of a three-sphere, i assume
such functions are the higher dimensional analogs of the spherical
harmonics Y_lm(theta,phi) ?
Any good books or articles where i might find:
1. the higher dimensional spherical harmonics,
2. and the various vector operators for four dimensions in spherical
coordinates?
I suspect that this is something a good math student should be able to
derive?
Re: Where to find spherical harmonics on a three-sphere.
Date: 1997/11/13
In article <6444b6$m70@r02n01.cac.psu.edu>
ale2@psu.edu (ale2) writes:
> The spherical harmonics Y_lm(theta,phi) show up a lot in quantum
> mechanics, for example the eigenfunctions for a particle constrained to
> move on the surface of a two-sphere. I am curious what the
> eigenfunctions look like for a particle which lives in four dimensions
> but is constrained to move on the surface of a three-sphere, i assume
> such functions are the higher dimensional analogs of the spherical
> harmonics Y_lm(theta,phi) ?
>
> Any good books or articles where i might find:
>
> 1. the higher dimensional spherical harmonics,
>
> 2. and the various vector operators for four dimensions in spherical
> coordinates?
>
For the answer to question #2 take a look at "Schaum's Outline Series
on Vector Analysis" by Murray R. Spiegel. As for question #1, I found a
short list of spherical harmonics for the 3-sphere in the following
book "Hyperspherical Harmonics, Applications in Quantum Theory" by John
Avery. For anyone interested they are (all constants are dropped)
Y_0,0,0 = constant
Y_1,0,0 = -u_4
Y_1,1,0 = iu_3
Y_1,1,1 = -i(u_1 + iu_2)
Y_1,1,-1 = i(u_1 - iu_2)
Y2,0,0 = 4(u_4)^2 - 1
Y2,1,0 = -iu_4u_3
Y2,1,1 = iu_4(u_1 + iu_2)
Y2,1,-1 = -iu_4(u_1 - iu_2)
Y2,2,0 = -2(u_3)^2 + (u_1)^2 + (u_2)^2
Y2,2,1 = u_3(u_1 + iu_2)
Y2,2,-1 = -u_3(u_1 - iu_2)
Y2,2,2 = -(u_1 + iu_2)^2
Y2,2,-2 = -(u_1 - iu_2)^2
in the above
u_1 = x_1/R
u_2 = x_2/R
u_3 = x_3/R
u_4 = x_4/R
and the 3-sphere is defined by those points which satisfy
(x_1)^2 + (x_2)^2 + (x_3)^2 + (x_4)^2 = R^2
Particle in a box versus in a 3-sphere.
Date: 1997/11/09
The wave-functions for a particle in a box go like
Sin(a*Pi*x/L)Sin(b*Pi*y/L)Sin(c*Pi*z/L)
The wave-functions for a particle constrained to move on a sphere go
like:
spherical harmonic Y_lm(theta,phi)
The wave-functions for a particle constrained to move on the surface of
a three-sphere go like???
spherical harmonic Y_lmn(theta,phi,chi) ???
Re: Particle in a box versus in a 3-sphere.
From: distler@golem.ph.utexas.edu (Jacques Distler)
Date: 1997/11/09
In article <644uve$6ea@newsstand.cit.cornell.edu>, bwr1@cornell.edu wrote:
> In article <64448f$m70@r02n01.cac.psu.edu>, ale2 <ale2@psu.edu> wrote:
> >The wave-functions for a particle in a box go like
> >
> > Sin(a*Pi*x/L)Sin(b*Pi*y/L)Sin(c*Pi*z/L)
> >
> >The wave-functions for a particle constrained to move on a sphere go
> >like
> >
> > spherical harmonic Y_lm(theta,phi)
> >
> >The wave-functions for a particle constrained to move on the surface of
> >a three-sphere go like???
> >
> > spherical harmonic Y_lmn(theta,phi,chi) ???
> >
> >Thanks for any help!
>
> If they're constrained to move on a 2-dimensional surface, why do you
> have three spatial coordinates?
Because he is talking about motion on a THREE-dimensional surface.
>
> What's the difference between "on a sphere" and "on the surface of a three-
> sphere?" They sound like the same thing to me. I'm not sure what you're
> asking.
"The sphere" is more properly refered to as the 2-sphere, the locus
x^2+y^2+z^2=1
in R^3. He is talking about the 3-sphere, which is the locus
x^2+y^2+z^2+w^2=1
in R^4.
To answer the qustion, we should remind ourselves where the spherical
harmonics came from. There, we were simply looking at representations of
SO(3), the group of rotations about the origin in R^3. The irreducible
representations of SO(3) are labeled by a nonnegative integer, l, and are
2l+1 dimensional.
These rotations take the 2-sphere into itself, so the generators of SO(3)
can be represented as differential operators in phi and theta
Exercise: Take the expression for the angular momentum generators in
cartesian coordinates,
L_x = -i(y d/dz - z d/dy)
L_y = -i(z d/dx - x d/dz)
L_z = -i(x d/dy - y d/dx)
and convert them to spherical coordinates. Note that the r-dependence
drops out, and these can be written as differential operators in theta and
phi.
Hint:
The easiest one is L_z = -i d/dphi
The Y_{lm} are simply a basis for the l'th irreducible representation,
consisting of eigenfunctions of L_z.
Now on to the 3-sphere. There we are looking for irreducible
representations of SO(4), the group of rotations about the origin in R^4.
This group has SIX generators, whose expressions are similar to the above,
with pairs of cartesian coordinates chosen from (x,y,z,w).
At this point, you could proceed by brute force, as above. Write your six
generators in hyperspherical coordinates (noting that the r-dependence
drops out), and try to find representations.
But it helps to know what you are aiming for. It helps to know that the
double-cover of SO(3) is SU(2). Representations of SU(2) are as above, but
with l either integral or half-integral We throw away the half-integral l
represetnations ot get representations of SO(3). Similarly, the
double-cover of
SO(4) is SU(2)xSU(2), So an irreducible representation is labeled by TWO
irreducible representations of SU(2):
Y_{l,m;l',m'}
To get a representation of SO(4), and not its double cover, we need to
demand that l and l' are either both integral or both half-integral.
Jacques Distler
--
PGP public key: http://golem.ph.utexas.edu/~distler/distler.asc
Re: What exactly is the spin of a particle?
Date: 1997/11/09
In article <MPG.ecd724252e4e385989682@news.mclink.it>
mc9350@mclink.it (Stefano Bianchi) writes:
> Ciao!
> I've always thought the spin as something linked to the momentum of a
> particle, and never cared too much about it. Now that I'm getting
> involved in Quantum Mechanics and becoming aware of its importance, I
> really need a more precise explanation of what it is.
For an interesting article on spin see "What is Spin" by Hans C.
Ohanian in American Journal of Physics, Volume 54 June 1986.
Link to thread of next article.
What operators acting on what objects give spinor field?
Date: 1997/12/03
Take the gradient of a scalar field and you get a vector field. What
object (if any) that is not a spinor field when operated on with some
operator gives a spinor field?
And while we are on the subject of spinors, does one have to use as a
basis two spinors which are in opposite directions, say the +z and -z
directions? Could one use +z and -x for example? When it comes to
vectors (say in 3-space) one can use any triplet of vectors as a basis
as long as they define a non-zero volume.
Re: What operators acting on what objects give spinor field?
Date: 1997/12/07
In article <66cn6f$blo@gap.cco.caltech.edu>
kevin@cco.caltech.edu (Kevin A. Scaldeferri) writes:
> In article <662i35$1i62@r02n01.cac.psu.edu>, ale2 <ale2@psu.edu> wrote:
> > Could one use +z and -x for example?
>
> One could, since |-,x> = (|+,z> - |-,-z>)/sqrt(2) so they span the
> space. However, it's unclear why you would want to do this since
> orthogonal bases are so much nicer and less of a headache.
Its unclear with me also %^)
but thank you, its nice to know and i didn't!
Re: What operators acting on what objects give spinor field?
Date: 1997/12/06
In article <662i35$1i62@r02n01.cac.psu.edu>
ale2@psu.edu (ale2) writes:
> Take the gradient of a scalar field and you get a vector field. What
> object (if any) that is not a spinor field when operated on with some
> operator gives a spinor field?
Well i came up with a pair of "objects" from which one can get a spinor
field but they are not very cute.
If one is given the electric field E and magnetic field B as functions
of space and time then one can use these fields to produce a spinor
field of space and time. If we consider a spinor as a flagpole with
flag then simply let the magnitude and orientation of the "flagpole" be
given by E X B (the cross product of E and B) and let the orientation
of the flag be given by either E or B.
Other examples would be appreciated!
Re: What is a spinor?
Date: 1997/12/04
In article <666nkb$1c28@r02n01.cac.psu.edu>
ale2@psu.edu (ale2) writes without thinking too much:
>
> A flagpole with a flag or a linearly polarized photon are examples. You
> need four numbers in general to describe a spinor,
The example above should be linearly polarized *plane* wave and not
linearly polarized photon which suggests some kind of localization
which would require more numbers.
Re: What is a spinor?
Date: 1997/12/04
In article <666d8g$93f@panix2.panix.com>
erg@panix.com (Edward Green) writes:
> Taking a break from the philosophy of science, however embarrassing...
>
> Well? In terms I am likely to understand, say in the spirit of "a
> tensor is a linear machine with some number of input slots".
>
> Are they members of some more general class of objects?
A flagpole with a flag or a linearly polarized photon are examples. You
need four numbers in general to describe a spinor,
three for the description of the directed line segment (two numbers
(angles) for direction and one number for its magnitude,
and one number for the orientation of the flag about the pole.