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Link to thread of next article. Particle in box --> quantized E, particle in ? --> quantized S . Date: 1996/05/22
If we limit a particle to move only inside a box then the particle will have quantized energy levels. What kind of "box" would give a particle quantized spin?
Re: Particle in box --> quantized E, particle in ? --> quantized S . Author: Matthew J. McIrvin <mmcirvin@world.std.com> Date: 1996/05/24Forum: sci.physics
In article <4o075s$fhl@news1.t1.usa.pipeline.com>, egreen@nyc.pipeline.com(Edward Green) wrote:
> 'ale2@psu.edu (ale2)' wrote: > > >If we limit a particle to move only inside a box then the particle will > >have quantized energy levels. What kind of "box" would give a particle > >quantized spin? > > Suppose a particle carried a little toroidal box around with it. The > boundary condition is now not reflection but matching the solution at 360 > degrees, and the ground state is now not a standing wave, but a > travelling wave which is two-fold degenerate in the absence of some > external perturbation. > Sound familiar?
Actually, the ground state would be a wave that doesn't travel at all. You could say, well, that's the spin zero representation, but if the different states correspond to different values of the spin, then you'll have trouble getting out 2n+1 - fold degeneracy for the nth energy level.
There's a better model that accomplishes all of this quite nicely. The particle *does* carry around a little box with it! It's the "box" of possible rotational orientations. And it's not a toroid, it's an SU(2), like John said. A somewhat different closed three- dimensional hypersurface (imagine waving a little flag around-- you can move the pole with two degrees of freedom and rotate the flag with a third--that's SO(3), and SU(2) is just that with the additional possibility of a funny minus sign for 360-degree rotations).
Actually, I think SU(2) is just the surface of a hypersphere. (SO(3) can't be because it's not simply connected.) -- Matt McIrvin http://world.std.com/~mmcirvin/
Re: Particle in box --> quantized E, particle in ? --> quantized S . Author: john baez <baez@guitar.ucr.edu> Date: 1996/05/31Forums:sci.physics, sci.math
In article <mmcirvin-2405961728140001@news.std.com> mmcirvin@world.std.com (Matthew J. McIrvin) writes:
>Actually, I think SU(2) is just the surface of a hypersphere. (SO(3) >can't be because it's not simply connected.)
Right, SU(2) is geometrically the same as what we mathematicians call S^3, or the 3-sphere --- which is the unit sphere in R^4.
We get the usual rotation group SO(3) by taking SU(2) and identifying each element x with the corresponding element -x. (Here "identifying" means "treating as if it were the same" or, more geometrically speaking, "gluing together".) Thus SO(3) is geometrically the same as the space you get by taking S^3 and identifying each point x with the antipodal point -x. More generally, mathematicians call the space obtained by taking S^n and identifying antipodal points "real projective n-space", or RP^n. It's fun to start by visualizing RP^2 in detail. RP^3 is not much harder: you can think of it as the 3-dimensional solid unit ball with antipodal points on its boundary identified. This gives the best way to visualize what's going on with the famous coffee cup trick.
The orientation of a body is described by a point in SO(3) so the study of a spinning body is mathematically the same as that of a free particle moving on SO(3). When we do quantum mechanics we merely need to remember to work with SU(2) instead of SO(3). So the quantum-theoretic study of spin is pretty much the same as the study of an abstract free particle particle moving around on S^3, satisfying Schrodinger's equation.
An adventure of superhero, Super Math-Physics Man! Date: 1996/04/25
One day, Super Math-Physics Man was feeling rather clever (no doubt inspired by other superheros such as Majorana, Dirac, Proca, et. al.) so he wrote (page 149, Annals of Math. 40 1939):
1. Origin And Characterization Of The Problem
"It is perhaps the most fundamental principle of quantum mechanics that the system of states forms a linear manifold, in which a unitary scalar product is defined. The states are generally represented in such a way that phi and constant multiples of phi represent the same physical state. It is possible, therefore, to normalize the wavefunction, i.e., to multiply it by a constant factor such that its scalar product with itself becomes 1. Then, only a constant factor of modulus 1, the so-called phase, will be left undetermined in the wave function. The linear character of the wave function is called the superposition principle. The square of the modulus of the unitary scalar product (psi,phi) of two normalized wave functions psi and phi is called the transition probability from the state psi into phi, or conversely. This is supposed to give the probability that an experiment performed on a system in the state phi, to see whether or not the state is psi, gives the result that it is psi. If there are two or more different experiments to decide this (e.g., essentially the same experiment, performed at different times) they are all supposed to give the same result, i.e., the transition probability has an invariant physical sense. The wave functions form a description of the physical state, not an invariant however, since the same state will be described in different coordinate systems by different wave functions. In order to put this into evidence, we shall affix an index to our wave functions, denoting the Lorentz frame of reference for which the wave functions given. Thus phi_l and phi_m represent the same state, but they are different functions. The first is the wave function of the state in the coordinate system l, the second in the coordinate system m. If phi_l = psi_m the state phi behaves in the coordinate system l exactly as psi behaves in the coordinate system m. If phi_l is given, all phi_m are determined up to a constant factor. Because of the invariance of the transition probability we have
and it can be shown that the aforementioned constraints in the phi_m can be chosen in such a way that the phi_m are obtained from the phi_l by a linear unitary operation, depending, of course, on l and m
The unitary operators D are determined by the physical content of the theory up to a constant factor again, which can depend on l and m..."
And so super Math-Physics Man continued and wrote what was to become a classic, maybe soon to be on the information highway?
If you want the Readers Digest version of the above classic try:
"Symmetry In Physics:Wigner's Legacy" in Physics Today, December 1995, page 46-50 or Quantum Field Theory, page 59- ,Lewis H. Ryder
Unitary Representations of the Poincare group for twits. Date: 1996/05/25
I'm trying to understand the main point of Wigner's famous 1939 paper but I'm a twit so I'm having some trouble. Apparently by combining Special Relativity and quantum mechanics spin appears (at the end of a long paper). There appears to be something wonderful going on here and I'm dying to find out what it is.
Can you recommend any material which is written about Wigner's paper whose audience might be twits like me?
Ideas for children, this is not physics, many apologies. Date: 1996/05/29
I was trying to get a 3 year old to draw over some simple shapes and letters i drew to get him interested in writing, he did not seem to care to follow over the shapes i drew. Then i tried something different. With a pen i drew a pair of parallel lines that curved this way and that to form a "race" course on a piece of paper. Then using a different color pen i "drove" around the race course trying not to crash into the boundaries. Then it was my boy's turn. He quickly got the idea and also drove around the track. Sometimes he would go off course, but if the "road was wide enough he could keep it on the track. We had several different tracks and sometimes we would have races in which some sound effects were added.
You might be able to have fun with this idea?
After reading some of the writing my daughter was to hand in i got disappointed with how bad it was and that got me to thinking how i could get her to have the best writing with the least work on her part (she doesn't want to be a writer). I am now trying to stress to her to READ what she writes. Now I know this seems so obvious but if she gets into the habit of just reading what she writes she will find some obvious mistakes. If she sees these mistakes enough times she will stop making them (hopeful thinking). If she continues this habit of studying what she writes she will find smaller and smaller mistakes? One would hope that by just studying what one writes one can answer the question is this writing doing a good job in communicating my ideas?
Its late, i hope i have done a fair job communicating my ideas, and hope these ideas were worth communicating.
Re: What to do with a smart kid. Date: 1996/10/02
In article <01bbb092$0b3420a0$4a033695@davea.eye.com> "David Arsenault" <davea@eye.com> writes:
> My son is 90% bored with his 5th grade math class. He picks up concepts > quite quickly, and doesn't easily forget them. He has little patience for > homework which simply practices the mechanics of computation.
Your child should not only know how to do the problems but he should be able to do them quickly as is required on a test. Does he?
> So, I'm > looking for some extra, fun math things to do with him.
A good library should have many books he will be interested in?
A neat series of books and not too costly, i think he'll love them:
The Joy of Mathematics, by Theoni Pappas
More The Joy of Mathematics, by Theoni Pappas
Re: What to do with a smart kid. Date: 1996/10/03
In article <DyoI0F.M33@world.std.com> jkenton@world.std.com (Jeff Kenton) writes:
> "David Arsenault" <davea@eye.com> writes: > > >My son is 90% bored with his 5th grade math class. He picks up concepts > >quite quickly, and doesn't easily forget them. He has little patience for > >homework which simply practices the mechanics of computation. So, I'm > >looking for some extra, fun math things to do with him. > > If he doesn't know algebra yet, I'd start him there. I did the same > with my daughters. Algebra is a basic requirement for most of the math > he'll see for several years, and for science as well. > > And, make sure it's fun. >
reminds of a game i sometimes play with my daughter till she gets bored, it goes like this (all work done in the chalk board of the mind or on paper):
I'm thinking of a number, add two to it and you get 4
I'm thinking of a number, double it and add two you get 12
I'm thinking of a number, subtract two from it and then double the result and you get 14
I'm thinking of a number, square it and add five and you get 30
you can teach them the trick to solve the problems, if they do the problem in "reverse" they will easily get the answer. They must not only reverse the order of the operations but they must use the "inverse" of the operation:
multiplication <--> division addition <--> subtraction square <--> square root
so to solve, I'm thinking of a number, double it and add two you get 12, you perform:
(12 - 2)/2 = 5
Re: Dummy with smart kid needs advice. Date: 1997/02/24
In article <01bc220f$51f9a3e0$77ceb7c7@default> "cambela" <cambela@ix.netcom.com> writes:
> hi, everyone. > I don't know where else to ask for advice, so I thought I'd try here. > My son is ten and has been asking me questions about what seems to > be the properties of heat which I believe falls into the physics > category.... > Can anyone here tell me if you know of any books for kids?
Take him on a regular basis to the local library. Find where the subject of interest is located and go through the shelves of books, he and *you* might find something interesting. There has got to be some good books he will enjoy even if he just looks at the pictures.
> It might be good for him to study and be good at something.
Being paid to do what you like has got to be the best!
> He's very smart in some areas but has a math block. and " ADD"
The math block may be something very simple. Give him the following test many times until you determine a pattern. Pick two numbers (from 1 to 9) and ask him to either add or multiply the two numbers. If he can't do this in a heartbeat he has work to do (i also have a 10 year old). If so, one way to practice (used with my daughter) is to have him make up problems, say 45*945 or 12345*9876, and have him solve these problems. He will get practice both adding and multiplying numbers. Allow him to use a times table and an addition table. Have him check his answers with a calculator. Have him do this 20 min. a day and i guarantee his math block will be busted. Work with him and reward his efforts. Good luck!
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