Talk:Hydrogen atom

Density function image
I've made a newer version of the density function image, i think it should replace the old one. Yey or ney? —Preceding unsigned comment added by PoorLeno (talk • contribs) 22:52, 16 August 2008 (UTC)

Relativistic effects of electron
The discussion of the speed of the electron stated that it "moved" at 1/100 the speed of light. My research shows that it's 1/10, and doesn't apply to the innermost shell. The primer I linked to includes the math. &mdash;DÅ‚ugosz
 * The speed of the electron is not a precisely definable quantity in a given energy state. However, the low-energy states have v/c on the order of the fine structure constant, which is closer in order of magnitude to 1/100 than 1/10.--76.81.164.27 05:59, 13 April 2007 (UTC)

Deleted "Schrodinger's Paradox" Stuff
I'm a physics major at the University of Rochester, and came here to look up an equation for the ionization energy for the hydrogen atom for my statistical mechanics class. To my surprise, I found a section in the hydrogen atom article entitled "Schrodinger's Paradox". I deleted the section because it is utter pseudoscience and is misleading on a number of grounds. It's also original research, which has no business being on wiki. It was also biased towards the idea being correct. —Preceding unsigned comment added by 128.151.144.60 (talk) 01:55, 7 April 2008 (UTC)
 * User:200.222.237.108 has been going around adding fringe science junk to various articles. You just caught a chunk of it. Sorry. S  B Harris 02:55, 7 April 2008 (UTC)

Wave function
Although the wave function may be correct for a particular definition of the Generalized Laguerre polynomials, the expression in the article (before my edit) was not if we use the definition in the Laguerre polynomials. I think we should be coherent with the other articles, so I have changed the expression for the wave function to use those polynomials.

(I'm new at editing wikipedia, so if I haven't done anything properly, I would like you to tell me, please. Thank you). —Preceding unsigned comment added by John_C_PI (talk • contribs) 19:26, 19 December 2005


 * It seems to me that now the Laguerre polynomials in Wiki is consistent with the one with (n+l)! instead of (n+l)!3.But I want someone to confirm this before I edit the page.Send me a message if anyone agrees or disagrees with me.--Netheril96 (talk) 12:23, 10 September 2010 (UTC)

GA on hold failed
Some minor things to adjust before the GA is awarded : Nothing was changed, the article will be failed. Lincher 13:44, 2 July 2006 (UTC)
 * Needs just a bit more references.
 * The Mathematical summary of eigenstates of hydrogen atom section is really tough to understand by itself, it needs more text surrounding it. Lincher 15:48, 23 June 2006 (UTC)

Wavefunction formula with (n+l)!^3
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:
 * $$ \psi_{nlm}(r,\theta,\phi) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- \rho / 2} \rho^{l} L_{n-l-1}^{2l+1}(\rho) \cdot Y_{l,m}(\theta, \phi ) $$
 * Polynomials:
 * $$L_\nu^\beta ( \rho ) = \sum_{m=0}^{\nu} (-1)^{m} \frac{(\nu + \beta)!}{m!(\nu - m)! (\beta + m)!} {\rho}^m \ ; \ \beta > -1$$
 * $$L_0^\beta ( \rho ) = 1$$
 * $$L_1^\beta ( \rho ) = - \rho + \beta +1$$
 * $$L_2^\beta ( \rho ) = \frac{\rho^2}{2} - ( \beta + 2 ) \rho + \frac{(\beta + 2)(\beta+1)}{2}$$
 * 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 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]:= $$R[n_{-},l_{-},r_{-}] := \sqrt {{\left ( \frac{2}{n a0} \right )}^3\frac{(n-l-1)!}{2n((n+l)!)} } \textrm{Exp}[- \frac{r}{n a0}] (\frac{2 r}{n a0})^{l} \textrm{LaguerreL}[n-l-1,2 l + 1, \frac{2 r}{n a0}]$$

Spherical part
 * In[2]:= $$Y[n_{-},l_{-},\Theta_{-},\phi_{-}] := \textrm{SphericalHarmonicY}[n,l,\Theta,\phi] $$

Full wavefunction
 * In[3]:= $$\Psi[n_{-},l_{-},m_{-},r_{-},\Theta_{-},\phi_{-}] := R[n,l,r] Y[l,m,\Theta,\phi] $$

Check normalization (3D volume integration in spherical coordinates, here for specific quantum numbers)
 * In[4]:= $$\textrm{Simplify}[\int_{0}^{\infty}(\int_{0}^{\pi}(\int_{0}^{2\pi}\textrm{Conjugate}[\Psi[n,l,m,r,\Theta,\phi]]\Psi[n,l,m,r,\Theta,\phi]r\textrm{Sin}[\Theta] d\phi)r d\theta)dr $$
 * $$ /.\{n->2,l->1,m->1\}, a0>0] $$
 * Out[4]= $$1$$

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 $$(n+l)!$$ 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 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 $$(n+l)!$$. I find the physics convention most clearly expressed in. 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 $$(n+l)!$$ seems to be well understood, but there seems to be another convention difference in which $$L_{n-l-1}^{2l+1}$$ is replaced by $$L_{n+l}^{2l+1}$$. This apparently regards a difference in how the Associated Laguerre polynomials can be defined as derivatives of the Laguerre polynomials. This is discussed in. One reference I've seen that seems to do this is 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)

poorly written
The level of this article is wildly inappropriate for the general reader. Much of the section "Features going beyond the Schrödinger solution" doesn't belong in this article. --76.81.164.27 05:56, 13 April 2007 (UTC)
 * Hear, hear! David F (talk) 03:19, 10 March 2013 (UTC)

Atomic hydrogen
Atomic hydrogen and Hydrogen atom do NOT have overlapping meanings. This article is junk. Was it written by a non-native English speaker?71.31.145.210 (talk) 18:41, 16 March 2012 (UTC)
 * They can be used interchangeably (say, for isolated hydrogen atoms), or can have different meanings (as in compounds), thus I disagree. Materialscientist (talk) 00:32, 17 March 2012 (UTC)

Eccess and Binding energy
"Eccess energy" and "binding energy" links to the same article, What is the diference? 80.24.186.207 22:42, 2 September 2007 (UTC)
 * The sign. 149.217.1.6 16:09, 27 October 2007 (UTC)


 * The explanation (the difference between an isotope's mass number and its actual mass in AMU (times c2, of course)) can be found at "Mass excess"; I have fixed the redirect. --DWIII (talk) 01:54, 28 September 2008 (UTC)

Wavefunction and normalization
As a follow-up to the discussion above about the different definitions of the associated Laguerre polynomials, I have come across the following problem: The radial part of the wavefunction as given in the article
 * $$ R(r) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- \rho / 2} \rho^{l} L_{n-l-1}^{2l+1}(\rho) $$

doesn't (quite) seem to be correctly normalized. The integral over r2|R|2 yields the value 2 for n = 1, 1 for n = 2, 2/3 for n = 3, 0.5 for n = 4, etc. Could someone double-check this? 149.217.1.6 16:08, 27 October 2007 (UTC)
 * Follow-up: I found the source of the discrepancy. When making the change of radial variable from $$r$$ to $$\rho$$, a factor $$dr/d\rho$$ is introduced in the integration. Hence,
 * $$\int r^{2} \left|R(\rho)\right|^{2} dr = \int r^{2} \left|R(\rho)\right|^{2} \frac{dr}{d\rho} d\rho$$
 * with
 * $$\frac{dr}{d\rho} = \frac{na_{0}}{2}$$.
 * With this additional factor, the integral over all space yields unity and the normalization condition holds. 149.217.1.6 20:18, 28 October 2007 (UTC)

H-1
Is H-1 a proper notation for the hydrogen atom? The only article in which I've seen it used is Big Bang nucleosynthesis. It is also not mentioned in the disambiguation page H-1. Should it be added to this page? --George100 (talk) 13:05, 12 July 2008 (UTC)
 * You usually write either "1H" or "hydrogen-1". BTW, why does it say "This article primarily concerns hydrogen-1", and why does hydrogen-1 redirect here? I see absolutely no reason why the stuff in this article can't apply to deuterium or tritium. -- Army1987 (t — c) 10:29, 30 October 2008 (UTC)

Incomplete and incorrect definition of j
I've chaged the definition of j: IIUC j is not an integer (it is an integer +/- 1/2). I've also added to the definition to clarify the meaning of 'total angular momentum'. I'm a bit rusty on this so please check! --Kevin Cowtan (talk) 12:11, 19 November 2008 (UTC)

electron spin -- schrodinger equation / pauli equation /dirac equation
there seems to be a bit of a muddle here concerning the electron spin and the schrodinger equation. the normal unmodified schrodinger equation describes  the behaviour of a spin zero particle, and as such will predict that a hydrogen atom is possessed of zero angular momentum in the ground state, contrary to what is observed. it is only when one steps up to the pauli a.k.a schrodinger-pauli equation that this defect is corrected. (or, of course, the fully relativistic dirac equation). this is not at all made clear in the present state of the article (march 2010), and will i fear need a deal of work to unravel. —Preceding unsigned comment added by 62.56.52.132 (talk) 22:51, 30 March 2010 (UTC)

Wrong diameter for illustration
The diameter is given as 2.4 Angstroms, or "twice the Bohr radius." In reality the correct diameter is about 1.1 Angstroms-- about half this. I realize these things are not exact, but 1.1 A is much closer than 2.4 Angstroms. The original artist and uploader apparently is no longer active. I wonder if anybody would like to re-do this illustration? S B Harris 04:58, 19 May 2010 (UTC)

Eigenstate 4,3,1, figure caption
There's a figure in the text showing a constant probability surface for the 4,3,1 state, the caption for which says, "The solid body contains 45% of the electron's probability." This is pretty ambiguous language, and it seems like it would be easy to misunderstand unless you already knew what it's supposed to mean; I mean, it contains "45% of the probability" for position measurements. I'm going to be bold and change it to say s/t like "3D illustration of the eigenstate $$\psi_{4,3,1}$$. Electrons in this state are 45% likely to be found within the solid body shown." I think this is a bit more precise, but it doesn't quite capture the fact that electrons aren't inside that space "45% of the time"; rather, position measurements are 45% likely to find them in that space. I haven't come up with a way to emphasize this subtle but key point in any kind of succinct way. I invite further suggestions for improved language. Flies 1 (talk) 16:27, 21 April 2012 (UTC)

Apparent editorial comments moved here from the article
In this edit I removed the following text which had just been added by an anon to the Quantum theoretical analysis section of the article:

"Emission spectrum of hydrogen. When excited, hydrogen gas gives off light in four distinct colours (spectral lines) in the visible spectrum, as well as a number of lines in the infra-red and ultra-violet. Bohr (and others) were aware of this and discovered orbitals happened to co-incide with wavelengths. From this exciting realization they began using electric force / motion functions and wave functions (Einstein) to explain the coincidence they saw in what they could only first imagine to be a classical orbit situation.

Before reading Schrödinger it's HIGHLY advised to check the acrticles concerning the spectrum of hydrogen and the articles concerning the simpler Bohr and Einstein models (which became quantum calculation) to see why these complexities became tried and relevant. (note also brownian motion and other experiments in the experiments listsing - these show allot too - and as well how often scientists move from bulk observation to particular causes and values underpinning)"

Wtmitchell (talk) (earlier Boracay Bill) 05:25, 16 July 2012 (UTC)

Dark matter
I removed the sentence about dark matter and dark energy from the lead paragraph because it was distracting. I left a link to the word "baryonic" in the preceding sentence, and that article mentions dark (non-baryonic) matter. If someone feels strongly about pointing out the difference between hydrogen and dark matter, I suggest it belongs somewhere other than the opening paragraph. - LesPaul75 talk 18:08, 14 January 2014 (UTC)

Wrong binding energy
The binding energy of the hydrogen atom is listed as "0.000± 0.0000 keV". It's actually about 13.6 eV, although I don't know the uncertainty. Comment by ‎128.111.18.19


 * That part of the infobox is called Nuclide data, and what is meant is the Nuclear binding energy to hold the nucleus together. Since there is only one nucleon, this is automatically zero. The atomic binding energy is 13.6 eV as you say, and is normally called the ionization energy.
 * I agree that this is confusing, and the entry really should be changed to say Nuclear binding energy. But I can't figure out how to make that change since the infobox only accepts labels that are on the preprogrammed template. Perhaps we should just delete that line from the infobox. Dirac66 (talk) 22:59, 12 August 2015 (UTC)

Error in Bohr-Sommerfeld Description
In the Bohr-Sommerfeld Model description, the symbol ε0 is referred to as "permeability." It should be "permittivity of free space." The symbol μ0 would be "permeability." — Preceding unsigned comment added by Nick0927 (talk • contribs) 02:34, 27 October 2016 (UTC)

Wavefunction
how can this be correct, $$ \sqrt {{\left ( \frac{2}{n a_0} \right )}^3 ...}$$ since a_0 is of dimension meters? Isn't there a factor like "r" missing like in $$ \rho = {2r \over {na_0}} $$? Ra-raisch (talk) 21:03, 4 December 2016 (UTC)


 * This factor implies that the wavefunction has units m-3/2 which is correct. The probability of finding the electron in a volume element dV is |Ψ|2dV which is dimensionless as required for a probability. And the integral of this probability over all space is 1, which is also dimensionless.


 * It would probably be helpful to explain this in the article, however. Dirac66 (talk) 22:14, 4 December 2016 (UTC)


 * is this cgs or SI? Ra-raisch (talk) 21:03, 18 December 2016 (UTC)
 * Well, I have written m-3/2 which assumes length in m so I have implied SI units. If you want to consider cgs with length units cm, then by a similar argument the units of Ψ would be cm-3/2. Dirac66 (talk) 21:23, 18 December 2016 (UTC)

Hydrogen atom mass - 1.007825 or 1.00794 ?
If you google 'hydrogen atom mass', Google gives result 1.00794 u at the top of page. That means most of the sites on internet mention 1.00794 u. I googled for 'hydrogen atom mass 1.007825'. Only 2 pages of results appeared and I think those sites are not reliable. Also in this article, 1.007825 is mentioned without any reference. Before removing this entry, I just want to confirm from you that I am not making any mistake. Thank you. — Preceding unsigned comment added by AbhiRiksh (talk • contribs) 18:47, 2 April 2018 (UTC)


 * Hydrogen is a mixture of 2 isotopes: 99.9885% H-1 and 0.000115 H-2 (terrestrial abundances). The mass of H-1 is 1.007825 u. The average mass of H is (1.007825 x 0.999885) + (2.014102 x 0.000115) = 1.007941 u. See data at Isotopes of hydrogen.
 * So the infobox is now technically correct because it specifies that the value 1.007825 is for the isotope H-1 (protium). However I think it would be clearer to give both values and explain the significance of each as well as the relationship between them. Dirac66 (talk) 23:47, 2 April 2018 (UTC)

Atomic hydrogen in relation to solvated electron and standard electrode potential
Some reactions of the solvated electron give atomic hydrogen and have electrochemical importance. I think it should be mentioned in article.--5.2.200.163 (talk) 15:42, 12 June 2018 (UTC)
 * Step 1: a strong reference supporting the assertion. --Smokefoot (talk) 18:26, 25 June 2018 (UTC)

Could it be that atomic hydrogen has identified identified in the past as nascent hydrogen?--5.2.200.163 (talk) 16:23, 25 June 2018 (UTC)

Reduced mass in solution to Schrödinger eq?
I've seen some texts saying that the µ letter refers to the reduced mass of the electron and the nucleus. I wonder why some texts (including Wikipedia) says that. In no moment during the formulation of the problem in Schrödinger equation appears the mass of the nucleus. The only reference of the nucleus is in the form of a Coulomb Potential. The letter µ refers to the electron mass, not to the reduced mass. — Preceding unsigned comment added by Heitorpb (talk • contribs) 00:55, 29 April 2019 (UTC)


 * Reduced mass is correct in this problem. Since the nucleus has a finite mass (even if it is much heavier the electron), it can move and has some kinetic energy which must be included in the total energy of the atom. Introducing the reduced mass allows the two-body problem (electron plus nucleus) to be transformed into an equivalent one-body problem, since the total (electron plus nuclear) kinetic energy is equal to an equivalent kinetic energy calculated with the reduced mass and the relative velocity. This is discussed in the article on reduced mass.Dirac66 (talk) 18:35, 2 May 2019 (UTC)

Unified Location and Treatment of Hydrogen Orbitals
Right now if someone wants information about hydrogen orbitals they may have to look at https://en.wikipedia.org/wiki/Hydrogen_atom#Wavefunction on "Hydrogen Atom" page and https://en.wikipedia.org/wiki/Hydrogen-like_atom#Non-relativistic_wavefunction_and_energy on "Hydrogen-Like Atom" page. Apart from already being inconvenient, it seems like these 2 pages sometimes end up differing in their normalization factor due to issues discussed above. It is a pain to keep both pages correct and consistent.

It would be convenient if there was one location where a reader could get all information about the hydrogen orbitals. In particular, I'd like to expand such a section with some helpful intuition and understandings to help readers predict and decipher the shapes of the hydrogen orbitals. However, I don't want to go through that effort until there's a clear dedicated location where this topic could be discussed.

My suggestion would be as follows: I don't think an entire article is needed for the hydrogen like atom. I think that article could be renamed to "Solutions to the Hydrogen Atom" or something like that without much change anywhere except the intro. Most of the article would then be about the mathematical solution to the Hydrogen atom (at varying levels of approximation) and there could be a small section on Hydrogen-like atoms (basically defining what they are and why they're useful). That article could then include the authoritative solution to the Hydrogen Schrodinger equation.

What do others think? Twistar48 (talk) 14:26, 27 October 2021 (UTC)


 * It is true that the articles Hydrogen-like atom could be combined with this one, following the procedure known in Wikipedia as merging. I would suggest retaining the name Hydrogen atom for the combined article, with one section called Hydrogen-like atom. We can explain that for hydrogen-like atoms, the Schrödinger equation and the orbitals contain a factor Z which is the nuclear charge. (There was one factor Z in the Wavefunction section, but I have just removed it since the rest of the section assumes Z = 1. If we carry out a merger, then the section on H-like atoms could include a factor Z consistently.)


 * The Hydrogen-like atom article now includes not only the Schrödinger equation but also a section on the Dirac equation which is not in the Hydrogen atom article. A merged article should include this section also. Dirac66 (talk) 18:20, 29 November 2021 (UTC)

Picture in info box should show the atom
I suggest to replace the picture in the info box (excerpt from table of nuclides) by something that shows the atom, e.g. file "Bohr model.jpg". The table of nuclides shows nuclear properties (and the isotopes separately); but here we are dealing with the atom. -- Wassermaus (talk) 21:15, 15 March 2022 (UTC)

Momentum representation of the Hydrogen atom is unphysical
The formulation of $$\varphi(p,\theta_p,\varphi_p)$$ involving a normalised product of a Gegenbauer polynomial and a spherical harmonic could actually be missleading. According to J.R. Lombardi (Chemical Physics 538 (2020) 110886, https://doi.org/10.1016/j.chemphys.2020.110886), the missconception stems from the fact that the momentum representation of a position-space wave function is its Fourier transform if and only if it is expressed in Cartesian coordinates. It is true that the Fourier transform of the spherical-coordinate wave function (as also performed in Bransden et al.) gives a mathematically correct expression, but these variables can not be interpreted as representations of momentum in quantum mechanics: they do not fulfill the standard commutation relation w.r.t. their position operator counterparts in curvlinear coordinates. As elaborated by Lombardi et al., the deWitt-transformation gives properly transformed and physically more reasonable expressions (where the spherical part no longer can be represented by spherical harmonics) with variables fulfilling the commutation relations. This missconception unfortunately is deeply rooted now in QM teaching and got replicated in quite some books, partly also because Fock, Pauling and Hylleraas fell for it. 2003:F4:8F20:4A00:FDA1:8FE7:ED46:8A30 (talk) 21:57, 25 July 2022 (UTC)