Talk:Atomic units/Archive 1

Archive 1

Atomic units related to magnetic quantities

In this article, there is a sentence:

" In atomic units, the Bohr magneton $\mu _{B}=1/2$, ..."

However, as described in Bohr magneton page, $\mu _{B}$ is expressed in S.I. units as

$\mu _{B}={{e\hbar } \over {2m_{e}}}$

and in Gaussian centimeter-gram-second units as

$\mu _{B}={{e\hbar } \over {2m_{e}c}}$.

If $e$, $\hbar$, and $m_{e}$ are set equal to unity in each of these expressions, the Bohr magneton is

$\mu _{B}=1/2$

in, S.I.-a.u., say, and

$\mu _{B}=1/(2c)$

in, Gaussian-a.u., say, and they have different values.

By virtue of the equality

$4\pi \epsilon _{0}=1$,

electric quantities have the same values in both S.I.-a.u. and Gauss-a.u.. For the magnetic quantities like $\mu _{B}$, however, this does not seem to be the case, and there seems to be a freedom or ambiguity which of S.I. or Gauss to be chosen.

Now, my question is, when a.u. is referred to, does it imply what I wrote as S.I.-a.u., as can be read from the part of article cited above? Is there a consensus, or a rule?

NorioTakemoto 15:14, 15 February 2006 (UTC)

A nice little problem in undergraduate physics! Your (and now my) perplexity purely concerns the cgs formula for μ_B, and I gather that cgs units are only of historical interest nowadays. That pesky little c... tsk-tsk. I will think about this.202.36.179.65 19:09, 9 April 2006 (UTC)

NIST defines the magnetic dipole atomic unit as 2 $\mu _{B}$. See http://physics.nist.gov/cgi-bin/cuu/Value?aumdm%7Csearch_for=atomic+unit However, then one would have to introduce another factor of c in the Maxwell equations if I am right - otherwise atomic dipoles would not have the same field in a.u. and in the SI system ( prefactor is ${\frac {\mu _{0}}{4\pi }}{\frac {m}{r^{3}}}$ for SI and ${\frac {1}{c}}{\frac {m}{r^{3}}}$ according to the formulation in the article. But if you actually calculate some numbers (with the a.u. for m as defined by NIST) this is off by a factor of c. If one calculates the field of a infinite straight wire (to get rid of the amiguity in the definition of the magnetic moment), we get $B={\frac {\mu _{0}I}{2\pi r}}$ in SI and would get $B={\frac {2I}{rc}}$ with the formulation as in the article. If we run some numbers (with the a.u. of current again defined the same way as NIST http://physics.nist.gov/cgi-bin/cuu/Value?aucur%7Csearch_for=atomic+unit ), it is also off by a factor of 1/c. So either one needs to define current and magnetic dipole moment with another factor of c or write Maxwell's equation with >math>c^{-2}</math> See also:http://en.wikipedia.org/wiki/Cgs_units#Various_extensions_of_the_CGS_system_to_electromagnetism http://en.wikipedia.org/wiki/Cgs_units#Electromagnetic_units_in_various_CGS_systems The article (and NIST) as of the unit definitions uses the a.u. equivalent to ESU-CGS, and therefore the prefactors to Maxwell's equations need to be $4\pi$ (correct in article), $-1$ (incorrect in the article) and $4\pi c^{-2}=4\pi \alpha ^{2}$ (also incorrect in the article). I will edit the article in a minute. 128.200.93.197 (talk) 18:47, 24 August 2009 (UTC)

Ultimately the problem is that under Gaussian conventions you get a magnetic field from the combination

${\frac {e}{a_{0}^{2}}}=1.72\times 10^{7}$ gauss

and under SI conventions you get a magnetic field from the combination

${\frac {\hbar }{ea_{0}^{2}}}=2.35\times 10^{9}$ gauss

So there's an ambiguity when you say "the magnetic field is 1 in atomic units", it could be 1.72E7 gauss or 2.35E9 gauss. I conclude that there are definitely two different "atomic units"s based on the definition in the article. So the question is, are both actually in use? Or only one? Can we find any sources that use one or the other, or that say explicitly that it's ambiguous? --Steve (talk) 15:53, 18 December 2009 (UTC) UPDATE: I checked, both are in use. Added this to the article. --Steve (talk) 23:34, 25 December 2009 (UTC)

The derived au table needs some attention

i might get to it myself at a later time. it would be good if au, Planck, Stoney, etc and all "natural units" were tied together into articles of consistent format. r b-j 02:36, 16 May 2006 (UTC)

I spent some time cleaning up both tables in the article. It didn't make sense to have Boltzmann and gravitational constants in the table. They were defined very differently from the other dimensional scales. Wigie 14:18, 18 May 2006 (UTC)

The Fundamental atomic units table contradicts BIPM

http://www.bipm.org/en/si/si_brochure/chapter4/4-1.html says that

" ... Similarly, in the a.u. system, any four of the five quantities charge, mass, action, length, and energy are taken as base quantities. The corresponding base units are the elementary charge e, electron mass me, action , Bohr radius (or bohr) a0, and Hartree energy (or hartree) Eh, respectively. In this system, time is again a derived quantity and the a.u. of time a derived unit, equal to the combination of units /Eh. Note that a0 = alpha/(4piRinfinity), where alpha is the fine-structure constant and Rinfinity is the Rydberg constant; and Eh = e2/(4piepsilon0a0) = 2Rhc0 = alpha2mec02, where epsilon0 is the electric constant and has an exact value in the SI. ..."

that is, energy is a base unit of the system. However for some reason, the article insists that the "electric constant" is a base unit, why is this? isn't it better to agree with BIPM. The article doesn't give any source for this choice.

--Alfredo.correa (talk) 06:57, 6 May 2010 (UTC)Alfredo

The units are all interrelated, because of equations like:

${\frac {e^{2}}{4\pi \epsilon _{0}a_{0}}}=E_{h}$

You define four of the units to be "1" and then prove that all the other units are "1". But it doesn't matter which ones are "fundamental" (or "base") and which ones are "derived", it just matters that they're all consistent. BIPM makes a different choice than this article does, but it's a pointless distinction anyway. The article doesn't make this very clear right now... --Steve (talk) 02:08, 11 May 2010 (UTC)

Comparison with Planck units

I made some changes here, but I'm still not happy with it. It seems that this section doesn't make its point very concisely, and in the process "hides" some interesting au values like the speed of light and the Bohr magneton. Wigie 14:23, 18 May 2006 (UTC)

Boltzmann's constant

The article says:

Finally, au normalize a unit of atomic energy to 1, while Planck units normalize to 1 Boltzmann's constant k, which relates energy and temperature.

However, if I understand the table of derived units correctly, atomic units also normalize Boltzmann's constant to 1. Am I missing something? Henning Makholm 15:01, 6 July 2006 (UTC)

Hi, same problem for me. Can someone give a reference to an external document where it is shown how the atomic unit for temperature is derived? As far as I can see, one has to define (not derive) ${\frac {E_{h}}{k_{B}}}=1a.u.$. I'm not convinced that this is a commonly used definition. Tovrstra 14:48, 19 March 2007 (UTC)

As far as I know, the "Hartree" atomic units system does not include a temperature definition at all. It was designed for computations involving single atoms and molecules, for which temperature isn't a meaningful concept. It was never intended to be a holistic, self-consistent set of units like MKS or CGS. Basically, when an ensemble of atoms or molecules is to be considered, the temperature definition used by everyone in the field is the standard Kelvin unit. Then, the Boltzmann constant is arrived at through the SI-defined Boltzmann constant and the conversion factor from Joules to Hartree, i.e., about (1.3807*10^-23 J K^-1)*(1 Hartree / 4.3597*10^-18 J) = 3.167 x 10^-6 Hartree / K . KeeYou Flib (talk) 16:05, 7 November 2019 (UTC)

The "atomic unit of temperature" was re-inserted recently after having been removed, apparently after the editor discovered it on the vCalc.com website because that was the reference given, with a horribly inaccurate value. The NIST site lists many atomic units (evidently Hartree atomic units), but it does not include an atomic unit of temperature. I made the re-included value more accurate and tagged it for citation, but I would like to remove it as unencyclopaedic (there is nothing motivating the choice of the Boltzmann constant as being set to 1 a.u. in this "system"). A Google search does show mention of the phrase in a few papers, but I see nothing notable to suggest that this is not just an ad hoc invention in each case. (Pinging George Church.) —Quondum 16:47, 12 December 2019 (UTC)

I have reverted the addition due to lack of a suitable reference to indicate that this is has ever been regarded as part of the system; I agree with Qflib above. Feel free to reopen this with a notable reference for this. —Quondum 00:52, 22 December 2019 (UTC)

Electric constant

It might be worth mentioning that, in SI units, the electric constant is an exactly defined (not approximately) unit. Replacing the common definition ${\frac {1}{4\pi \epsilon _{0}}}$ with the equivalent ${\frac {\mu _{0}c^{2}}{4\pi }}$ simplifies to $10^{-}7c^{2}$ exactly, or 8987551787.3681764 kg m^3 s^-2 C^-2.
--65.202.227.50 (talk) 15:46, 18 February 2010 (UTC)mjd

The exact definition of ε0 is important if you're talking about how SI works, but if you're talking about atomic units I don't think it matters too much. The expression ${\frac {1}{4\pi \epsilon _{0}}}$ is very common and familiar, it would be very confusing to people if we stopped using that expression and instead used the uncommon expression ${\frac {\mu _{0}c^{2}}{4\pi }}$. Since they're the same thing anyway we should use the more common and easy-to-recognize expression I think. :-) --Steve (talk) 00:10, 19 February 2010 (UTC)

Another Atomic units system in use?

In the Hartree system the Coulomb constant is set to be 1, is there a system where instead the speed of light is set to 1 and the other settings are the same?

It would be very similar to Hartree, but the equation for the energy of the wave function $\psi$ of an electron in a H-like-atom (nucleus with Z protons and only one electron in the shell) would be:

${\hat {H}}\psi =-{\frac {1}{2}}\Delta \psi +\alpha {\frac {Z}{r}}\cdot \psi$

with the fine-structure constant $\alpha =1/137.03..$ --MrWithFly (talk) 07:32, 10 October 2020 (UTC)

I think you may be thinking of Planck units, which are mentioned in the article. Unless I've missed something? KeeYou Flib (talk) 17:22, 19 February 2021 (UTC)

Repeated citations

I have just realized that many of the citations in the article are repeated citations of the same article, which should be cleaned up. If anyone wants to get to that before I can, please feel free! Qflib, aka KeeYou Flib (talk) 15:00, 19 July 2022 (UTC)

Nondimensionalization

Hi, I have some questions about recent additions to the page by an unregistered IP editor 172.82.46.195 under "Non-relativistic quantum mechanics in atomic units". I wanted to get a conversation going before making any changes, especially with the editor in question.

Could someone please explain to me why it might be correct to refer to the quantities and equations as nondimensionalized? When I look at these equations expressed in atomic units, I think of dimensions: energies in Hartree, lengths in Bohr/a.u. of length, and so on, rather than the equation being nondimensionalized (as in the usual way of non-dimensionalizing the quantum harmonic oscillator, where the point is to make the solution simpler by scaling various terms by the mass/force constant/frequency), since there are no conversion factors involved in bringing the solutions of an nondimensionalized equation back to a dimensionalized form - just inverse application of the scaling definitions, In other words, when atomic units are used $m_{e}$ is still there in the equation, but has a value of exactly 1.00000... atomic units and so need not be written explicitly. But of course the Schrodinger equation for the hydrogen molecule includes both nuclear masses as explicit parameters whether or not atomic units are used.

Overall I think of the Hartree units as an (incomplete) system of units, with base units which correspond to the electron's physical properties. Thus the eigenvalues of the time-independent Schrodinger equation for any given system are in atomic units of energy, which has a known and specific conversion factor to SI.

Thoughts anyone? Qflib, aka KeeYou Flib (talk) 16:33, 19 July 2022 (UTC)

Thanks for pinging me. I'm happy to discuss. The underlying reason for my presenting it in this way is that there is a lot of confusion of units, mainly relating to the distinction between systems of quantities (as opposed to systems of units), resulting in nonsense such as "The molar mass of a substance is the mass of one mole of that substance." This results in imprecise and confused statements in the articles here. A mathematically consistent framework is straightforward to construct once the underlying idea is understood.

There are essentially two distinct common conventions when it comes to natural units. They are different because their systems of quantities are distinct, one being the same as used with SI units (dimensional) and one being without units (nondimensional).

I'm not sure why you would regard Hartree units as an incomplete system: four dimensions is all that are needed for describing quantities in physics above the subatomic level. Relating Hartree units to my statements above, one must be clear which of the two conventions (and hence systems of quantities) one is using from the outset. The section on nondimensionalization uses the other convention when compared to the rest of the article, and tries to be explicit about this. 172.82.46.195 (talk) 21:47, 19 July 2022 (UTC)

On review of my answer, I think an example may be called for. We can say that the electron mass is m_e ≈ 9.109×10⁻³¹ kg ≈ 5.486×10⁻⁴ Da ≈ 1 a.u. of mass. Given any mass m, we can define a new quantity m′ ≝ m/m_e, and call it, to give it a name, the mass-factor corresponding to m. In the nondimensionalized system of quantities, we use only each such quantity-factor, but typically take a shortcut and just use the original quantity name and symbol (e.g. "mass" and m instead of "mass-factor" and m′ in this example). In the nondimensionalized equations, we should really have a prime or similar indication on each quantity to distinguish it from the corresponding dimensional quantity. For correctness, would also write m_e′ = e′ = ħ′ = 4πε₀′ = 1. We even have a symbol to denote the corresponds-to relationship: m′ ≘ m. 172.82.46.195 (talk) 23:26, 19 July 2022 (UTC)

That's very clear- thanks very much for taking the time to respond so thoroughly. As far as using primes or whatever, I think we should be sticking to common usage - it's an encyclopedia, after all!

Let me follow up with an example of what I mean. In my comments I obliquely referred to the case of the hydrogen molecule. For simplicity, consider ${\ce {H_2^+}}$, which has a single electron. Numbering the the protons as A and B, and the single electron as 1, the nonrelativistic Hamiltonian is

${\hat {H}}=-{{{\hbar ^{2}} \over {2m_{\text{e}}}}\nabla _{1}^{2}}-{{{\hbar ^{2}} \over {2m_{\text{p}}}}\nabla _{A}^{2}}-{{{\hbar ^{2}} \over {2m_{\text{p}}}}\nabla _{B}^{2}}-{1 \over {4\pi \epsilon _{0}}}{{e^{2}} \over {r_{1A}}}-{1 \over {4\pi \epsilon _{0}}}{{e^{2}} \over {r_{1B}}}+{1 \over {4\pi \epsilon _{0}}}{{e^{2}} \over {r_{AB}}},$

where all quantities are in any consistent unit system (MKS, cgs, or what have you). In Hartree atomic units, we would express this Hamiltonian as

${\hat {H}}=-{{{1} \over {2}}\nabla _{1}^{2}}-{{{1} \over {2m_{\text{p}}}}\nabla _{A}^{2}}-{{{1} \over {2m_{\text{p}}}}\nabla _{B}^{2}}-{{1} \over {r_{1A}}}-{{1} \over {r_{1B}}}+{{1} \over {r_{AB}}},$

with $r_{1A}$ the distance from electron 1 to proton A, and so forth. As we see, the proton mass appears as a symbol in this expression, and has to be assigned a value which is consistent with the unit system in order to work out. I guess from your POV you'd want to put a prime on the proton mass...right?

Incidentally my comment about this not being a "complete unit system" is that it doesn't have a definition for things like temperature, and luminous intensity, which are part of SI. Qflib, aka KeeYou Flib (talk) 15:28, 20 July 2022 (UTC)

One can continue using a quantity (such as temperature) which is outside the mechanics context in the "usual" units, SI or like. On the other hand, one rarely encounters the need to use temperature in atomic physics calculations. And when one does, it's usually in the form of $k_{B}T$ (dimension of energy), in which case $k_{B}$ is assumed unity as well and the temperature counted in the energy units, eV/27.2. Evgeny (talk) 15:56, 20 July 2022 (UTC)

Absolutely. See my comment above on this page under "Boltzmann constant." Qflib, aka KeeYou Flib (talk) 16:19, 20 July 2022 (UTC)

In the "nondimensionalized" version of any equation (using the choice referred to), the general effect is to reinterpret every dimensional quantity as a nondimensionalized corresponding quantity. One then simply drops the four constants as being equal to 1. So in the example given here, m_p remains in the formula (by convention, one does not generally use the prime: it is helpful for clarity and explanation, though), but has been reinterpreted as the dimensionless value 1836.15. I'm not sure this leads to any suggestion for a clarification in the article, though. The nondimensionalization is something seized on by some WP editors as a "simplification", but my preference would be to remove this section entirely as being a side-track in the encyclopedic context. In my mind, the only point that could be made reasonably is that these units lend themselves to this shortcut in this context, whereas SI units do not.

On temperature, etc., SI is an "overcomplete" system, in the sense that it defines three dimensions that are purely technical: they can be fully described in terms of the other four, since there is direct correspondence between average kinetic energy per degree of freedom and temperature, for example, so the temperature scale is not describing a "new physical dimension". For temperature in the atomic system, one just uses this corresponding quantity, which arguably is temperature. One also just counts atoms (or whatever), rather than number of moles, and one is describing exactly the same thing, though if preferred one can extend the system with an SI-like temperature dimension as Evgeny suggests.

I don't see anything contentious or unclear here. Is there something in the article that could be improved/clarified? My own suggestion would be to reduce the section under discussion to a mere mention of nondimensionalization or remove it altogether. I did not do so since I expected pushback. 172.82.46.195 (talk) 20:21, 20 July 2022 (UTC)

On the one hand, now that you've explained it, it's perfectly clear what is meant. On the other hand, I did need the explanation, so I'd guess that others might too - sounds like you don't think so. I'm okay with leaving it as is, because of course, it might just be me. Qflib, aka KeeYou Flib (talk) 21:02, 20 July 2022 (UTC)

Ideally, it should not need explanation: when anyone needs explanation, rewording might be called for. I've adjusted it a bit, with the aim of framing the intent a bit better at the start. Though of course, it is difficult to judge how successful this is if one already knows what is to be achieved. 172.82.46.195 (talk) 22:31, 20 July 2022 (UTC)

Looks good to me, nice edit. Qflib, aka KeeYou Flib (talk) 15:55, 21 July 2022 (UTC)