## If the angular momenta commute!!!!

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blacksl
Posts: 25
Joined: Wed Dec 22, 2010 4:51 am

### If the angular momenta commute!!!!

If the angular momenta about three axes of rotation in a central potential did commute, how many quantum numbers would the
system have overall?

HappyQuark
Posts: 762
Joined: Thu Apr 16, 2009 2:08 am

### Re: If the angular momenta commute!!!!

blacksl wrote:If the angular momenta about three axes of rotation in a central potential did commute, how many quantum numbers would the
system have overall?

As a general rule in this forum and many other physics forums, you ought to at least try the problem on your own and then explain your reasoning. Once you've done that we can offer up advice and point you in the right direction.

CarlBrannen
Posts: 381
Joined: Mon May 24, 2010 11:34 pm

### Re: If the angular momenta commute!!!!

And besides, it's one of those ill-posed problems that's kind of difficult to answer. It's like "if your aunt had testicles would she be your uncle?"

blacksl
Posts: 25
Joined: Wed Dec 22, 2010 4:51 am

### Re: If the angular momenta commute!!!!

what I think is,
if angular momenta commute then system can simultaneously have 3 well defined angular momenta around x,y and z.
also Lx, Ly, Lz commute with L^2.

so what I think is total # of quantum numbers is 7 (1+3+3).

CarlBrannen
Posts: 381
Joined: Mon May 24, 2010 11:34 pm

### Re: If the angular momenta commute!!!!

There ought to be another quantum number for the energy, unless it's one of the rare cases where there's only one energy / radius. For example, how many quantum numbers does the 1/r central potential have?

And by the way, when $L_x,L_y,L_z$ commute with each other, I don't see why $L^2$ should be listed as a quantum number. It's eigenvalue is given by $l_x^2+l_y^2+l_z^2$ which is not independent of your other three eigenvalues. So that way I only count three, but I'd still include the energy. And maybe spin, if it's an electron going round. So something like 3, 4, or 5.

All this reminds me of

No Radial Excitations in Low Energy QCD I & II, Tamar Friedmann
http://arxiv.org/abs/0910.2231
http://arxiv.org/abs/0910.2229

P.S. Still no acceptances. If someone knows of a department who needs a hot shot electronics engineer / grad student, PGRE = 990, please send them my way. I'm easy to find on the web. Works cheap. House trained. Doesn't shed much.

blacksl
Posts: 25
Joined: Wed Dec 22, 2010 4:51 am

### Re: If the angular momenta commute!!!!

Because [Lx,L^2]=0, [Ly,L^2]=0, [Ly,L^2]=0. this means Lx,L^2 share same eigen function. if angular momentum commute with each other then there are 3 quantum number for that and 3 degenerate quantum numbers due to [Lx,L^2]=0

Im confuse!!

CarlBrannen
Posts: 381
Joined: Mon May 24, 2010 11:34 pm

### Re: If the angular momenta commute!!!!

blacksl wrote:Because [Lx,L^2]=0, [Ly,L^2]=0, [Ly,L^2]=0. this means Lx,L^2 share same eigen function. if angular momentum commute with each other then there are 3 quantum number for that and 3 degenerate quantum numbers due to [Lx,L^2]=0

Im confuse!!

One only uses as many quantum numbers as are needed to describe a state. For example, the usual angular momentum quantum numbers are $L_z,L^2$. These are a "maximal set of commuting observables".

But the point in this is that $L_z$ and $L^2$ are independent. Knowing one does not tell you what the other is.

If you allow non independent quantum numbers, you might as well include an infinite number of them. Here's an infinite set of commuting operators (or observables): $L_z,L_z^2,L_z^3,...,L_z^n,...$ (As an exercise, verify that these operators all commute and compute their eigenvalues given the eigenvalue for $L_z$.)

-----------------------------------
P.S. Still no acceptances. If someone knows of a department who needs a hot shot electronics engineer / grad student, PGRE = 990, please send them my way. I'm easy to find on the web. Works cheap. House trained. Doesn't shed much. Theory or experiment.

blacksl
Posts: 25
Joined: Wed Dec 22, 2010 4:51 am

### Re: If the angular momenta commute!!!!

Thanks "CarlBrannen"
it make sense...

I'm wondering why could not find any school even-though you have 990 for physics GRE!!!....
By the way are you looking for transferring ?
I hope you will get good one!!!

Good Luck

CarlBrannen
Posts: 381
Joined: Mon May 24, 2010 11:34 pm

### Re: If the angular momenta commute!!!!

blacksl wrote:I'm wondering why could not find any school even-though you have 990 for physics GRE!!!....
By the way are you looking for transferring ?
I hope you will get good one!!!

Good Luck

It's cause I always shoot for the top.
At the time I was applying, I was thinking "why should I go to grad school at any but the very top university?", but having geared up for school (I hit the books before the GRE and I'm continuing to hit them in preparation for grad school), I'm finding I'm enjoying the studying. So I'm more enthusiastic about going than I was when I was applying.

On the other hand, I've got some opportunities in scrapping available out at Moses Lake. The metals have gone crazy. Scrap steel is selling at \$385 per ton. My buddy says he's making \$2000 per day cutting apart locomotives. I wouldn't call myself an expert, but I can use a cutting torch, work cheap, and don't get into trouble with drugs or alcohol (which, surprisingly, is actually an important detail for this sort of work). If I do that for six months I could show up at grad school self funded for the first year.

Another buddy is setting up a gold mine in Idaho and has invited me out to help. That is awfully attractive, it is beautiful country. With my other buddy, I managed to set him up with a pretty good deal on an hydraulic excavator and a loader. The excavator is a Komatsu P400. It weighs a bit over 50 tons and can dig about 40 feet below the track level. I'll post a photo when the deal completes. Can't wait to try it out.

Also, a much better place to ask physics questions like this is probably the new Physics Stack Exchange. For example:

"Will a ball slide down a lumpy hill over the same path it rolls down the hill?"
Yes, see:
http://physics.stackexchange.com/questions/728/

-----------------------------------
P.S. Still no acceptances. If someone knows of a department who needs a hot shot electronics engineer / grad student, PGRE = 990, please send them my way. I'm easy to find on the web. Works cheap. House trained. Doesn't shed much. Theory or experiment. Can repair and run heavy equipment, weld, etc.