Talk:Particle in a spherically symmetric potential

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1 Extra factor
2 Normalization 3D harmonic oscillator functions.
3 Cleanup template
4 Query re 4D symmetry of coulomb potential energy

[edit] Extra factor

Where does the extra $\sqrt{\gamma}$ come from in the radial solution? Can that factor be lumped into the normalization constant, or is there a pedagogical reason why that particular dependence should be indicated?

As an aside, with the exception of that square root factor I was able to make a correspondence between the final expression for the radial equation with those in Bransden & Joachain (2000). They use (what I believe is) $n = {\lambda \over 2} - {3 \over 2}$ and $\alpha^2 = \gamma \;\,$ , and $n_r = {n -l \over 2}$ . --HappyCamper 04:01, 30 May 2007 (UTC)

This factor arose naturally in the derivation that I did on paper. I indeed considered dropping it, as the solution is unnormalized anyway. If you find it unnecessary, please go ahead and remove it. It is good that you confirmed the result. I have heard of Bransden and Joachain, but never looked at it. I had no book handy that gave the result, so I had to trust myself. BTW, do B&J give a normalization constant? If so, could you add it to here? Thanks.--P.wormer 07:34, 30 May 2007 (UTC)

B+J doesn't have the normalization constant...I found a book which gave the normalization constant, but I really think this should be double checked. I was not able to find more than one source to verify the results. Journal papers seem to solve the radial equation for a D-dimensional oscillator, but I have not been able to find a correspondence between these results and the textbooks even after careful substitutions. Some authors use a different definition of Laguerre polynomials. (It's actually quite difficult to find this normalization constant in a textbook; most gloss over it.) Also, I figured out where the initial discrepancy was between

r l

and

r l + 1

. It comes about if you do a change of variables to solve for the associated wavefunctions of the 1-D S.E. - also, interesting that a lot of textbooks give the normalization constant in terms of the confluent hypergeometric function. --HappyCamper 22:33, 30 May 2007 (UTC)

I commented out temporarily the normalization factor because I am pretty sure it is wrong.

(n - 2 l)!

can be undefined (argument of factorial can be negative) and I believe that the denominator should contain

π 1 / 2

not

π 1 / 3

. --P.wormer 10:07, 31 May 2007 (UTC)

I rederived the normalization constant [using Wikipedia as a source for the formulas :-) ] and came up with the equation that is now in the article. Question now is: who is correct, Mathews and Venkatesan or myself? (Or did HC make a copy/TeX error)? BTW is this derivation against Wiki:NOR? --P.wormer 11:31, 31 May 2007 (UTC)

I wouldn't call this original research - we are simply Wikipedians who care about what we do. We're doing our homework properly, double checking, triple checking, cross-referencing, to make sure content here is 100% factual! :-) To put this in historical context, about a year ago, there was a similar conversation regarding the derivation of the entropy of some probability distribution. I am absolutely certain that there was a factor of

π 1 / 3

in that book, because my first thought was "Hmm...that's the first time I've seen that number crop up!". I think you are right to change it to 1/2, that seems more reasonable. However, look at Equation 28 of this other paper - the factor of pi is missing!

No, it is not, it is in the Gamma function. --P.wormer 08:18, 1 June 2007 (UTC)

So, I am not even sure if this pi should show up or not. I cited M+V because it was the only book I found after going through around 30 textbooks that actually had a formula printed out. They give the radial equation as

$R_{nl}(r) = N_{nl} \cdot r^l e^{(-\frac{\alpha^2 r^2}{2})} L^{l+{1 \over 2}}_{\frac{1}{2}(n+l+1)}(\alpha^2 r^2)$ with normalization $N_{nl} = \left [ \frac{2^{n+l+2} \,\alpha^{2l+3}\,[{1 \over 2}(n+l)]!\;(n-2l)!}{(n+l+1)! \;\pi^{1 \over 3}} \right ]^{1 \over 2} .$

They use $\alpha = \sqrt{\frac{m \omega}{\hbar}}$ so it means that $\alpha = \sqrt{\gamma}$ . They also use this definition of associated Laguerre polynomials (pg. 132):

$\rho \frac{d^2}{d\rho^2}L^p_q + (p + 1 - \rho) \frac{d}{d\rho}L^p_q +(q-p) L^p_q = 0$ which satisfy (pg. 133) $\int_0^\infty e^{-\rho} \rho^{p+1} [L^p_q(\rho)]^2 d\rho = \frac{(2q-p+1)(q!)^3}{(q-p)!}$

This is the old-fashioned definition of Laguerres as also given by Margenau & Murphy (p.128). Nowadays the subscript on L gives the degree (highest power) of the polynomial. Subtract the superscript from the subscript to go from the old to the new definition. Thus the Laguerres of Mathew and V. are equal to $L^{(l+\frac{1}{2})}_{\frac{1}{2}(n+l+1) - l -\frac{1}{2}} = L^{(l+\frac{1}{2})}_{\frac{1}{2}(n-l)}$ . This settles one discrepancy. However, the normalization integral you quote is of no use in the present problem. In the normalization a half-integral power of ρ arises. Thus, the integral results in a Gamma function with half-integral argument, which in turn gives the factor π^1/2.--P.wormer 08:13, 1 June 2007 (UTC)

Thanks for the explanation. Yes, this normalization was not useful. --HappyCamper 23:38, 1 June 2007 (UTC)

I think this discrepancy with the factor of (n-2l)! comes from this different definition. It should be simple to check, but I have to run off for now. I'm going to do another literature search...there must be a more authoritative source for this normalization constant. After all, the equation must have been first solved by somebody, and it would be priceless to quote this original paper here. --HappyCamper 18:30, 31 May 2007 (UTC)

The definition of n is given unambiguously by the energy expression. What energy do Mathews and V. have?--P.wormer 08:13, 1 June 2007 (UTC)
I didn't copy this down, and I don't have the book anymore. They have this equation:

$\xi \frac{d^2 K}{d\xi^2} + (l + \frac{3}{2} - \xi) \frac{dK}{d\xi} + \frac{1}{4} (\lambda - 3 - 2l) K = 0$ and $\lambda ={2m_0 E \over \hbar^2 \alpha^2} = \frac{2E}{\hbar \omega}$

I take it that this equation is from Bransden and J. not from Mathews and V.? Because it differs from the one you quoted earlier? --P.wormer 07:50, 2 June 2007 (UTC)

Equation 18 of this paper looks promising: [1] --HappyCamper 20:26, 31 May 2007 (UTC)

I had Eq. (18) along the way, but did not like the double factorial, which I replaced to arrive at the result in the article. Note the π^-1/4 in this equation!--P.wormer 06:57, 1 June 2007 (UTC)

When you get a chance, could you take a look and see if you agree with the results? I added an extra step, because of the similarity to the functional form of the normalization constant for a D dimensional isotropic oscillator. --HappyCamper 01:01, 2 June 2007 (UTC)

Will start new section. This is getting messy--P.wormer 07:50, 2 June 2007 (UTC)

[edit] Normalization 3D harmonic oscillator functions.

What went before:

After some discussion HappyCamper and I derived a normalization constant that differed from the one in Mathews, P.M.; K. Venkatesan (1978). A textbook of quantum mechanics. McGraw Hill.. See the version of the article of 02:57, 2 June 2007 and talk under heading "Extra factor". --P.wormer 08:21, 2 June 2007 (UTC)

Here is the link. --HappyCamper 17:31, 2 June 2007 (UTC)

Hallo HappyCamper, it is good that you checked the derivations. However, I feel that the article is slowly getting clogged up with (too many) derivations. To avoid this I skipped earlier several of my pencil steps, also in the transformation from the SE (Schrödinger) to the Laguerre equation. To avoid loosing overview I propose the presentation technique used in several math articles: We give the SE and, without any ado, its normalized solution. Then we give a fairly full derivation under the [hide] HTML/CSS macro, so that readers can move this calculus out of sight. I am willing to do this, but not in the next couple of days, because I will be busy outside Wiki. I will then also check your formulas (most of which I have on paper). Finally, I find a double factorial of an even number fairly silly, for: $(2 n)!! = 2 n n!$ . --P.wormer 08:21, 2 June 2007 (UTC)

I agree more or less with your thinking. Too many derivation details at the moment, and it detracts from the essentials. I'm tempted to move this derivation to a new page say quantum isotropic oscillator where we could potentially expand on D-dimensional isotropic oscillators, and also make some correspondences between the 2D system and the Morse oscillator. Then, on this page, we can just put the essentials: Radial equation, normalized wavefunction, general properties of the states, et cetera. I like the double factorial, simply because it's easier to remember. As for the equation hiding, if we move the material to a new article, then it may not be essential. It's probably easier to do this, but I would like to experiment with the HTML/CSS here as well. Which math article has these? I have some things to do outside of the Wiki as well. I'll wait for you to double check your notes. In the meantime, I'm going to see if I can find another textbook that has this constant written out. --HappyCamper 17:31, 2 June 2007 (UTC)

I tried very hard to find an article with the [hide] macro, but could not find it anymore. I remember that I looked at it and that it was based on the css visibility: hidden attribute. Anyway, I would prefer to have extensive derivations hidden by default and to have a show macro to pop them up. For the time being we may give the final result (normalized radial wavefunction) first and then give the derivation. --P.wormer 09:41, 11 June 2007 (UTC)

With regard to moving contents: the 3D oscillator fits very nicely in the present article because it starts from the equation for u(r) that was derived for all central symmetric potentials. The 2D oscillator is almost the same (also Laguerre for the radial part). It could go in here. I don't know about higher than 3D (except, of course, Cartesian plus 2nd quantization), although I could imagine that hyperspherical harmonics could be useful for 4D.--P.wormer 09:41, 11 June 2007 (UTC)

I performed the restructuring (separation of result from derivation). I used "my" form of the normalization constant in presenting the result, because I feel that neither the Gamma function nor the double factorial are widely known (especially not among chemists).--P.wormer 12:35, 11 June 2007 (UTC)

I couldn't find anything useful either. But I'm sure I've seen them somewhere too. Anyway, I resorted to asking on the help desk: [2]. Hopefully we will get a useful response. The higher-than-3d will be for another day. Thanks for the restructuring - the normalization constant looks great! --HappyCamper 02:46, 12 June 2007 (UTC)

[edit] Cleanup template

Who put the cleanup template in the article and why?--P.wormer 12:41, 11 June 2007 (UTC)

Looks like the cleanup tag was placed in December 2005...someone probably noticed that the image links were dead. The images were deleted almost a year earlier on January 2005. [3]. I'm not sure how it ended up being at the top. I think we can remove it though. A nice thing to do for this article is to add pictures, but I don't think I will get to that anytime soon. --HappyCamper 02:32, 12 June 2007 (UTC)

[edit] Query re 4D symmetry of coulomb potential energy

Peter Atkins's popular (but comparatively deep) science book, Galileo's Finger: The Ten Great Ideas of Science (2003) mentions the coulomb potential energy as having 4D symmetry, which he says is responsible for the degeneracy (or near degeneracy?) of e.g. 2s and 2p orbitals in hydrogenic atoms, but not for those with more than one electron; indeed, he provides a beautiful and convincing illustration of how a spherical s-orbital can be rotated into a two-lobed p-orbital!). But nowhere does he seem to clarify what the 4th component of the symmetry is [e.g. I don't see how it could be a symmetry in time as well as space]. I've searched the web on "coulomb 4D symmetry" etc., but found nothing that seems to clarify the basis for the 4D symmetry. (The closest ref might be the J D Jackson reference at the end of the main Gauge Fixing page, but it doesn't mention 4D symmetry.) Would the "Particle in a spherically symmetrical potential" page be a suitable place to open up this aspect in a little more depth for enquirers at my level (PhD in chemistry but my maths and group theory are not at the level of a graduate physicist). If this is not the best place in the wikipedia, perhaps someone could kindly point me to a more apt location to ask. Thanks. 163.119.193.40 20:37, 14 August 2007 (UTC)

Hmm...try at the Mathematics reference desk. They often like to try and answer these sorts of interesting questions. Hope this helps, and do check back frequently here as well, since others might have other ideas for you to pursue as well. (Wikipedians generally don't answer queries via e-mail, so I removed it from your post.) HTH. --HappyCamper 03:32, 15 August 2007 (UTC)

Thanks. And I wasn't expecting an answer via email - I just couldn't see how to put an equivalent of your "Happy Camper" in, so I signed it with my email address. 163.119.193.40 11:46, 15 August 2007 (UTC)

No problem. Since you're editing anonymously, Wikipedia uses the string of 4 numbers as your signature in place of a username. --HappyCamper 16:07, 15 August 2007 (UTC)