Talk:Hydrogen-like atom

Normalization?

For the non-relativistic wavefunction's radial term, Griffiths (and other sources) provide the normalization factor:

${\sqrt {{\left({\frac {2Z}{na_{\mu }}}\right)}^{3}{\frac {(n-\ell -1)!}{2n{[(n+\ell )!]^{3}}}}}}$

On the page, the denominator of the second fraction in the radical is just $2n(n+\ell )!$, not $2n[(n+\ell )!]^{3}$. Is this a typo?

Bohr Radius

a0 as defined in the article is not the Bohr radius. The Bohr radius does not contain Z or mu. Perhaps we should not call this variable a0 here, because a0 usually means the Bohr radius. My reference (Bransden and Joachain, Intro to QM) defines a quantity a_mu, which is the same as a0 in the article but does not contain Z, and calls it the modified Bohr radius. Should we use that instead? We'd have to replace a0 with a_mu/Z everywhere, then. Pfalstad 21:42, 8 January 2006 (UTC)

Did you mean?:

$a_{\mu }={\frac {4\pi \epsilon _{0}\hbar ^{2}}{\mu e^{2}}}$

I had seen this definition too, and probably it would fit better here because of the definition used in Bohr radius. To be consistent, I agree that we would better do as you say and change that. (Anyway, note that $a_{\mu }\approx a_{0}$ almost exactly, for a single electron orbiting a proton, so this should be said in the article). --John C PI 22:45, 10 January 2006 (UTC)

I agree. I would think that a0 should be replaced with a_μ/Z and that the formula for "a" should not include Z. I would make the changes myself, but I don't really have much experience with using the math formatting, plus I am not familiar with the given general form of the wavefunction equation.--GregRM 21:32, 3 March 2006 (UTC)

I went through and tried to make the changes. I would appreciate it if at least one or two people could check my edit to make sure I got everything correctly. Thanks.--GregRM 13:19, 25 June 2006 (UTC)

I don't know if your editing did this, but the energy term (E_n) MUST depend on Z. So either a_u does(and it doesn't seem to above), or somone flubbed during editing. —Preceding unsigned comment added by 128.189.132.69 (talk) 18:39, 27 October 2007 (UTC)

Energy?

Griffiths did a much better job in is QM book - even though he didn't handle the Z <> 1 case directly, it was much easier to walk around the formulae there than to look it up here. This article is badly organized and incomplete. Why are the energies not mentioned? Shinobu 21:45, 11 September 2006 (UTC)

Is the fact that the nucleus charge number, Z, is not included in the energy of the associated lepton, a mistake?

Mathematical analysis

section on Schrödinger equation in a spherically symmetric potential moved to a more appropriate place, under "Schrödinger equation" article. Dan Gluck 12:35, 22 May 2007 (UTC)

• (On 25 May 2007 moved from there to this article by HappyCamper).--P.wormer 14:43, 30 May 2007 (UTC)

New Edition

This article has been edited with relevant information about the energy of hydrogenic systems in non relativistic and relativistic frames. The sections on spherical harmonics are moved to relevant sections.--Raghunathan,02 June 2007

I created 1s Slater-type function and reverted this page back to its current state: [1]. There are a number of reasons why this page should not be replaced with the new version - to say the least, the current article structure on Wikipedia treats this page as being subordinate to hydrogen atom (e.g. see link), where most of the analytic derivations are expected to be carried out. --HappyCamper 01:44, 12 June 2007 (UTC)

From the lead

It seemed out of place at the end of the lead - putting it here for discussion -

In quantum chemical calculations hydrogen-like atomic orbitals do not play an important role, because they are not complete (they do not span all of one-electron Hilbert space).<ref>H. Shull and P.-O. Löwdin, J. Chem. Phys. vol. '''23''', p. 1362 (1955).</ref>

sbandrews (t) 20:47, 16 July 2007 (UTC)

I wrote that because of the following sentence which was in the lead before I changed it:

Orbitals of multi-electron atoms are qualitatively similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, numerical approximations must be used. Atomic orbitals are often expanded in a basis set of Slater-type orbitals which are orbitals of hydrogen-like atoms with arbitrary nuclear charge Z.

This sentence states that Slater type orbitals (STOs) are hydrogen-like orbitals (quod non, STOs can have a fixed exponent in the radial exponential and are then complete. An essential property of a hydrogen orbital is that its radial function depends on its principal quantum number, which is the cause of the incompleteness of hydrogen functions). Since STOs were introduced here as an expansion basis and STOs were identified with hydrogen-like orbitals it was strongly suggested that hydrogen orbitals are suitable as an expansion basis. Indeed, I have heard this erroneous statement more often. I wanted to make clear once and for all that hydrogen orbitals are NOT suitable as an expansion basis. As an additional remark: for me an "approximate numerical" method is a pleonasm, there is no such thing as an exact numerical method (at least when it is executed on a computer with a finite word length).--P.wormer 06:36, 17 July 2007 (UTC)

Agreed on the pleonasm, I wanted to put the approximate in there for the more general reader who may not appreciate the meaning of numerical methods in this context, I have re-worded a bit. As for the incompleteness I see the point you are making, by all means reinsert the text, perhaps some re-wording is needed, I will take another look, sbandrews (t) 07:07, 17 July 2007 (UTC)

Atom? I think you mean ion!

This page should be renamed, as the only hydrogen-like atom is hydrogen. I've never heard anyone refer to hydrogen-like atoms, only hydrogen-like ions, as this is in fact what any isoelectronic nucleus bar hydrogen is. To stipulate: ANY atom that has less/more electrons than protons is an ion, not an atom, and ALL "atoms" under discussion here share this property, hence all atoms under discussion are in fact ions, and the pae should be renamed as such, for accuracy! —Preceding unsigned comment added by 82.42.92.193 (talk) 15:02, 20 April 2010 (UTC)

I don't agree. Any atom with a single electron outside a singly charged core can be (and is) considered a hydrogen-like atom. Xxanthippe (talk) 23:00, 20 April 2010 (UTC).

Yes, I am reasonably sure this is what this odd term means. I have never heard the term "hydrogen-like ion" in physics, as this would refer only to a proton. Too bad we can't identify experts in their subject on WP to separate them from anyone who happens to have an opinion on a matter. While ad hominem arguments are invalid, establishing the background of an editor might help resolve silly differences of opinion about actual facts. Note that it is also true that a hydrogen-like atom can also happen to be an ion. The term is helpful in explanations of quantum mechanics, when a net charge on a tiny object isn't relevant to the explanation. David Spector (talk) 12:37, 23 September 2021 (UTC)

Nope, I think you'll find that any atom with a number of electrons not equal to a number of protons is an ion. Granolanifa (talk) 21:50, 13 May 2010 (UTC)

Well, that is certainly true. But it is irrelevant to this topic. Any atom with a net electronic charge is an ion, but not a hydrogen-like atom. David Spector (talk) 19:22, 20 December 2022 (UTC)

Non-relativistic wavefunctions

The radial part of the wavefunction has an extra $2Zr/(na_{\mu })$. Not easy to see why on dimensional grounds, as the factor is dimensionless. One needs to recall the behavior at small $r$. This is known to be $r^{l}$. In the typical derivation the substitution $R_{nl}=u/r$. With this substitution $u$ behaves like $r^{l+1}$ at small $r$.