The wavefuction formulas on Hydrogen atom and Hydrogen-like atom were recently changed ( and , respectively) to have (n+l)!^3 instead of (n+l)!. I have come across several instances with the (n+l)^3 form (e.g. ); this also seems to contain the (n+l)!^3 version, but the generalized Laguerre polynomials have subscripts of n+l, instead of n-l-1 as they are in Wikipedia's articles. I am guessing that maybe separate definitions of generalized Laguerre polynomials are being used, as suggested by a comment above by User:John C PI (cf. this edit)? This page has the (n+l)! version (I am assuming the use of (n+1)! is a typo) with Laguerre subscript of n-l-1. I tried a quick check in my head for n = 2, l = 1; based on Eq. 33 and 36 at , it seems that the use of (n+l)! with the n-l-1 degree generalized Laguerre would give the (presumably) correct result provided here, whereas the n+l degree version would result in a polynomial in r of at least degree 3. (Also, the use of (n+l)!^3 instead of (n+l)! would seem to give a different constant muliplier than provided in the previous link.) I am going to revert the changes based on my limited investigation into this issue...if anyone is able to confirm the validity of my assessment or clarify the seemingly contradictory results that I found, that would be great.--GregRM 20:39, 29 January 2007 (UTC)
- Yes, this is true. The problem is that different sources use different definitions for laguerre polynomials, and we expect Wikipedia to be consistent. In fact, when I studied the quantum physics subject (I'm a student of physics), it was very confusing that the two professors we had used different definitions! Anyway, the reversion you did is correct if we want to be consistent with the definitions in the Generalized Laguerre Polynomials article.
- I don't remember which recognised books use which definition, and which is more widespread, since my references are my professor's notes, which are correct. But at the time I first dealt with this for some reason I thought the definition in the Generalized Laguerre polynomials article was more appropiate (at least, in this last article there is no history of doubt, and this is a good signal).
- To clarify further doubts, this is a correct group of formulas and polynomials:
- Wavefunction:
![{\displaystyle \psi _{nlm}(r,\theta ,\phi )={\sqrt {{\left({\frac {2}{na_{0}}}\right)}^{3}{\frac {(n-l-1)!}{2n[(n+l)!]}}}}e^{-\rho /2}\rho ^{l}L_{n-l-1}^{2l+1}(\rho )\cdot Y_{l,m}(\theta ,\phi )}](//wikimedia.org/api/rest_v1/media/math/render/svg/0c69d0a09aeb218e5e747d2938e58602e7661ece)
- Polynomials:
- ;\ \beta >-1}
![{\displaystyle L_{\nu }^{\beta }(\rho )=\sum _{m=0}^{\nu }(-1)^{m}{\frac {(\nu +\beta )!}{m!(\nu -m)!(\beta +m)!}}{\rho }^{m}\ ;\ \beta >-1}](//wikimedia.org/api/rest_v1/media/math/render/svg/abeacf27772f9552495c7e3c341cd9470631026f)
![{\displaystyle L_{0}^{\beta }(\rho )=1}](//wikimedia.org/api/rest_v1/media/math/render/svg/ef92f0c398fd06c26adac01a6e965cf3bb52a165)
![{\displaystyle L_{1}^{\beta }(\rho )=-\rho +\beta +1}](//wikimedia.org/api/rest_v1/media/math/render/svg/0e6833ab0c12998dc85180b280edcc66739ff8e0)
![{\displaystyle L_{2}^{\beta }(\rho )={\frac {\rho ^{2}}{2}}-(\beta +2)\rho +{\frac {(\beta +2)(\beta +1)}{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/2807119d365c206bcaab7c2e05e057327b2b16b6)
- I hope this makes it a little more clear. John C PI 23:03, 29 January 2007 (UTC)
- I just corrected this error (again) from another (well-intentioned) anonymous editor. We need to keep an eye on this one. —Preceding unsigned comment added by Stephenedie (talk • contribs) 02:22, 20 January 2010 (UTC)
I edited the subscript (degree) of the Laguerre polynomial appearing in the wave function and cited a couple of references that use this convention (n-l-1). I am happy to see the degree displayed as n+l, but please give a reference for this convention if you change it back. Cheers! — Preceding unsigned comment added by Micah.prange (talk • contribs) 15:26, 17 August 2011 (UTC)
The formula for the radial part is incorrect. If the generalized Laguerre polynomials are defined as in the corresponding article, than the correct factor should be (n+l)!, not [(n+l)!]^3. Just check the normalization condition. — Preceding unsigned comment added by 129.100.61.12 (talk) 17:05, 19 September 2011 (UTC)
A simple check in math program, e.g. Mathematica, confirms the correct normalization factor with (n+l)! in the denominator (not cubed). Here is the output:
Radial part
- In[1]:=
![{\displaystyle R[n_{-},l_{-},r_{-}]:={\sqrt {{\left({\frac {2}{na0}}\right)}^{3}{\frac {(n-l-1)!}{2n((n+l)!)}}}}{\textrm {Exp}}[-{\frac {r}{na0}}]({\frac {2r}{na0}})^{l}{\textrm {LaguerreL}}[n-l-1,2l+1,{\frac {2r}{na0}}]}](//wikimedia.org/api/rest_v1/media/math/render/svg/946c4295049e61f815dec5dabe4f56cea80f39d5)
Spherical part
- In[2]:=
![{\displaystyle Y[n_{-},l_{-},\Theta _{-},\phi _{-}]:={\textrm {SphericalHarmonicY}}[n,l,\Theta ,\phi ]}](//wikimedia.org/api/rest_v1/media/math/render/svg/c337cb3524db3682890b1b373ba4893c88d01af8)
Full wavefunction
- In[3]:=
![{\displaystyle \Psi [n_{-},l_{-},m_{-},r_{-},\Theta _{-},\phi _{-}]:=R[n,l,r]Y[l,m,\Theta ,\phi ]}](//wikimedia.org/api/rest_v1/media/math/render/svg/acef1ee3daa2902becc7c07a8642529e08f7d86e)
Check normalization (3D volume integration in spherical coordinates, here for specific quantum numbers)
- In[4]:=
![{\displaystyle /.\{n->2,l->1,m->1\},a0>0]}](//wikimedia.org/api/rest_v1/media/math/render/svg/da8a49f0c13b84e928e1f636c1c570a441fff335)
- Out[4]=
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
Thomas.fernholz (talk) 14:52, 7 May 2012 (UTC)
- In that case, I'm getting rid of the square brackets because I really thought someone removed the "^3" as vandalism, since the square brackets are completely redundant anyways. --Freiddie (talk) 23:54, 7 May 2012 (UTC)
The issue is that the article cites Griffith as the source of the equation but Griffith uses a different definition of the Laguerre polynomials, his are a factor of
larger. So people keep coming along seeing it's different from Griffith and adding the ^3 thinking it is a typo. I removed the reference to Griffith and added a note below pointing out this difference in definition. Now there is no reference for the equation, but none of the text books I have looked at use this definition of the Laguerre polynomials and give a statement of the general hydrogen wave function, even Messiah as far as I can tell. Timothyduignan (talk) 03:38, 4 December 2012 (UTC)
Apparently that same error has found its way back into the article. Was just thrown off by it, took an hour until I figured it out. Corrected it. 88.215.115.26 (talk) 10:20, 12 May 2018 (UTC)
The definition of the nomalized energy eigenfunction given here agrees with that given in other Wikipedia articles and in common textbooks. The normalization factor used here differs from that given in Messiah, who replaces the present (n+l)! by [(n+l)!]^3 (Ch. 11, Sec. 6). The definition of the associated Laguerre polynomials used by Messiah (Appendix B2) differs from that given by eg. Abramowitz and Stegun by a factor of (n+l)!, which I believe agrees with that given in other Wikipedia articles. I have numerically checked the orthonormality of the present definition of the wave functions together with the definition of the associated Laguerre polynomials given by Abramowitz and Stegun. I suggest that the statement saying that the present definition is consistent with Messiah be removed.
— Preceding unsigned comment added by 49.180.73.107 (talk) 05:17, 28 May 2019 (UTC)
- I agree with this comment that Messiah differs from the convention in the article. See my comment below. If my comment below "sticks" I can clean up the text regarding the Messiah reference. Twistar48 (talk) 11:12, 16 November 2021 (UTC)
I've added a section at Laguerre polynomials#Physicist scaling convention that I hope will help clarify the discussion regarding the appropriate normalization factor. It seems that throughout Wikipedia (and the numerical computation and math worlds) the convention in Abramowitz is taken. In the physics literature the convention is taken which differs by a factor of
. I find the physics convention most clearly expressed in [1]. This gives us a hard time on Wikipedia because we want to stay consistent with the "standard" convention, but we cannot use that convention and directly cite the physics literature without giving an explicitly caveat as to why the formula on Wikipedia differs from that found in the referred textbooks. It seems to me the best strategy is to be as clear as possible about the different convention choices and which is taken where.
The convention difference regarding the factor of
seems to be well understood, but there seems to be another convention difference in which
is replaced by
. This apparently regards a difference in how the Associated Laguerre polynomials can be defined as derivatives of the Laguerre polynomials. This is discussed in [2]. One reference I've seen that seems to do this is [3] which is of note because this reference is directly cited by Griffiths in the context of Laguerre polynomial conventions. Twistar48 (talk) 11:12, 16 November 2021 (UTC)
References
Merzbacher, Eugen (1998). Quantum mechanics (3rd ed.). New York: Wiley. ISBN 0471887021.
Boas, Mary L. (2006). Mathematical methods in the physical sciences (3rd ed.). Hoboken, NJ: Wiley. ISBN 9780471198260.
Schiff, Leonard I. (1968). Quantum mechanics (3d ed.). New York: McGraw-Hill. ISBN 0070856435.