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What is a hydrogen-like or hydrogenic atom?
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Michael Romeo
What is a hydrogen-like or hydrogenic atom?
Hydrogen-like atoms are atoms with a single electron "orbiting" a nucleus which has more than one nucleon. As @Xerxes pointed out in a comment, you can in principle have a nucleus made up of particles other than protons and neutrons (nucleons). Positronium might be an extreme example of this.
Wikipedia actually has an entry about Hydrogen-like atoms which goes somewhat beyond what you asked.
Hydrogen-like atoms are atoms with a single electron "orbiting" a nucleus which has more than one nucleon. As @Xerxes pointed out in a comment, you can in principle have a nucleus made up of particles other than protons and neutrons (nucleons). Positronium might be an extreme example of this.
Wikipedia actually has an entry about Hydrogen-like atoms which goes somewhat beyond what you asked.
There are technical issues with this answer: The core of a hydrogen-like atom need not be made of nucleons. Also, deuterium has more than one nucleon and it is not hydrogen-like, it is hydrogen.More
Hydrogen-like ions are ions that possess only one electron, just like a hydrogen atom.
The fact that these ions have only one ion in the outermost shell makes it simpler to analyze their radii and energies, as a simple electrostatic model can be used to describe them. Species having multiple electrons are difficult to study, and are beyond the scope of the Bohr model. This is because the inter-electronic replusions are hard to account for in electrodynamic interactions that makes up the atom's bound system.
Hydrogen-like ions are ions that possess only one electron, just like a hydrogen atom.
The fact that these ions have only one ion in the outermost shell makes it simpler to analyze their radii and energies, as a simple electrostatic model can be used to describe them. Species having multiple electrons are difficult to study, and are beyond the scope of the Bohr model. This is because the inter-electronic replusions are hard to account for in electrodynamic interactions that makes up the atom's bound system.
I dont see a single mentioning of Bohrs model neither in the OP nor in the post linked there. And as Bohrs model doesnt work correctly even for hydrogen atom (e.g. ground-state angular momentum), I see no reason to use it at all, at least in the context of current question.More
On top of the previous comment, the Rydberg formula was only empirically determined, and is therefore not tied to any particular theory. It is not derived from the Bohr model; in fact, the Bohr model was developed to rationalise it. However, the use of the Bohr model is no longer necessary, as the QM model accounts for the Rydberg formula perfectly well. So this is not "a question about something regarding the Bohr model".More
). The difficulty arises from the electron-electron repulsions.More
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A hydrogen-like atom (or ion) is simply any particle with a nucleus and one electron.
That should be sufficient to answer the question at hand, but I thought I should say a bit more, as some of these answers are potentially confusing.
The historical reason why the Rydberg formula only works for hydrogen-like atoms is because it was originally formulated to explain the spectral lines of hydrogen. It was never intended to explain the spectra of multi-electron atoms.
The physical reason, though, is because the Rydberg formula uses energy levels that only depend on the principal quantum number $n$, which has to be a positive integer:
and nowadays we know that this is only true for hydrogen-like atoms;$^*$ the energy levels of multi-electron atoms depend on both $n$ and $l$.$^\dagger$
The $n$-dependency was later successfully rationalised by the Bohr model, but saying that "the Rydberg formula only works for hydrogen-like atoms because the Bohr model only works for them" is misleading and misses the point, as:
This implies that the Rydberg formula was derived from the Bohr model, which is not true; it was merely empirically determined, and the formula predated the Bohr model by 25 years.
The Bohr model simply does not work for hydrogen-like atoms. The fact that it reproduces the Rydberg formula should merely be considered a serendipity; Bohr arrived at the correct result by the wrong method.
It does not lend any real insight into the proper reason why the Rydberg formula does not apply for helium, etc. (which I briefly mentioned above).
$^*$ In fact, the energy levels of hydrogen are not only dependent on $n$ (due to various small effects such as – but not limited to – spin-orbit coupling, and hyperfine splitting). Wikipedia has a good overview of the topic here and most QM textbooks have a chapter on the hydrogen atom, where they discuss these perturbations to the Hamiltonian and their effects on the energies. Not surprisingly, the inability to explain this was one of the failures of the Bohr model.
$^\dagger$ Of course, there is also a series of approximations here. The energy levels of multi-electron atoms are only approximately described by sums of orbital energies, so the transition energies are only approximately equal to a difference in energy between two orbitals.
A hydrogen-like atom (or ion) is simply any particle with a nucleus and one electron.
That should be sufficient to answer the question at hand, but I thought I should say a bit more, as some of these answers are potentially confusing.
The historical reason why the Rydberg formula only works for hydrogen-like atoms is because it was originally formulated to explain the spectral lines of hydrogen. It was never intended to explain the spectra of multi-electron atoms.
The physical reason, though, is because the Rydberg formula uses energy levels that only depend on the principal quantum number $n$, which has to be a positive integer:
and nowadays we know that this is only true for hydrogen-like atoms;$^*$ the energy levels of multi-electron atoms depend on both $n$ and $l$.$^\dagger$
The $n$-dependency was later successfully rationalised by the Bohr model, but saying that "the Rydberg formula only works for hydrogen-like atoms because the Bohr model only works for them" is misleading and misses the point, as:
This implies that the Rydberg formula was derived from the Bohr model, which is not true; it was merely empirically determined, and the formula predated the Bohr model by 25 years.
The Bohr model simply does not work for hydrogen-like atoms. The fact that it reproduces the Rydberg formula should merely be considered a serendipity; Bohr arrived at the correct result by the wrong method.
It does not lend any real insight into the proper reason why the Rydberg formula does not apply for helium, etc. (which I briefly mentioned above).
$^*$ In fact, the energy levels of hydrogen are not only dependent on $n$ (due to various small effects such as – but not limited to – spin-orbit coupling, and hyperfine splitting). Wikipedia has a good overview of the topic here and most QM textbooks have a chapter on the hydrogen atom, where they discuss these perturbations to the Hamiltonian and their effects on the energies. Not surprisingly, the inability to explain this was one of the failures of the Bohr model.
$^\dagger$ Of course, there is also a series of approximations here. The energy levels of multi-electron atoms are only approximately described by sums of orbital energies, so the transition energies are only approximately equal to a difference in energy between two orbitals.
I think one key aspect that is overlooked in these replies is the Born-Oppenheimer approximation. The nucleus of any atom can be approximated as a point with very few relativistic correction due to its mass. An electron on the other hand is essentially massless (comparatively speaking. I know it is still a fermion and has mass). Once you put more than one electron in any orbital those relativistic corrections have consequences. So ''hydrogen-like'' means any atom that does not need Pauli or Dirac's insight to explain the deviations in the spectra i.e. the bohr model.
I think one key aspect that is overlooked in these replies is the Born-Oppenheimer approximation. The nucleus of any atom can be approximated as a point with very few relativistic correction due to its mass. An electron on the other hand is essentially massless (comparatively speaking. I know it is still a fermion and has mass). Once you put more than one electron in any orbital those relativistic corrections have consequences. So ''hydrogen-like'' means any atom that does not need Pauli or Dirac's insight to explain the deviations in the spectra i.e. the bohr model.
When you solve the H-atom problem the "conventional way" you first use the BO approximation and then fix the proton to the origin. When you solve it in the way I have scetched you dont apply the BO and you dont fix the proton. Rather you decompose the problem in relative motion of the two wrt centre of mass and the motion of the centre of mass (which you dont care about since thats just plane wave). the relative motion problem is completely identical to the conventinal H-atom except you obtain a different effective mass, hence energy levels.More
That is a good point. I guess I thought that Dirac applied relativity to explain the Pauli exclusion principle. In the case of the hydrogen atom you can simply switch to the center of mass. Is that the same as normal coordinates? I thought that is what the BO approximation did to limit the 3N degrees of freedom thus reducing the amount of parameters needed to approximate more complicated systems i.e anything past a two body problem.More
I am not sure if you dont confuse some things. For example you can straight forward analytically solve the hydrogen atom considering the electrons and protons mass. You simply switch to center of mass coordinates. Levels change by roughly 1+(1/(1+1836)). Thats actually beyond BO. Relativity you need for high "nuclear" effective charges $Z$. Thats quite different things.More
Hydrogen-like atoms are atoms with a single electron "orbiting" a nucleus which has more than one nucleon. As @Xerxes pointed out in a comment, you can in principle have a nucleus made up of particles other than protons and neutrons (nucleons). Positronium might be an extreme example of this.
Wikipedia actually has an entry about Hydrogen-like atoms which goes somewhat beyond what you asked.
Hydrogen-like atoms are atoms with a single electron "orbiting" a nucleus which has more than one nucleon. As @Xerxes pointed out in a comment, you can in principle have a nucleus made up of particles other than protons and neutrons (nucleons). Positronium might be an extreme example of this.
Wikipedia actually has an entry about Hydrogen-like atoms which goes somewhat beyond what you asked.
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Single electron systems are known as hydrogenic / hydrogen like species like He+, Li2+, Be3+ etc.
Single electron systems are known as hydrogenic / hydrogen like species like He+, Li2+, Be3+ etc.
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Hydrogen-like ions are ions that possess only one electron, just like a hydrogen atom.
The fact that these ions have only one ion in the outermost shell makes it simpler to analyze their radii and energies, as a simple electrostatic model can be used to describe them. Species having multiple electrons are difficult to study, and are beyond the scope of the Bohr model. This is because the inter-electronic replusions are hard to account for in electrodynamic interactions that makes up the atom's bound system.
Hydrogen-like ions are ions that possess only one electron, just like a hydrogen atom.
The fact that these ions have only one ion in the outermost shell makes it simpler to analyze their radii and energies, as a simple electrostatic model can be used to describe them. Species having multiple electrons are difficult to study, and are beyond the scope of the Bohr model. This is because the inter-electronic replusions are hard to account for in electrodynamic interactions that makes up the atom's bound system.
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A hydrogen-like atom (or ion) is simply any particle with a nucleus and one electron.
That should be sufficient to answer the question at hand, but I thought I should say a bit more, as some of these answers are potentially confusing.
The historical reason why the Rydberg formula only works for hydrogen-like atoms is because it was originally formulated to explain the spectral lines of hydrogen. It was never intended to explain the spectra of multi-electron atoms.
The physical reason, though, is because the Rydberg formula uses energy levels that only depend on the principal quantum number $n$, which has to be a positive integer:
$$\bar{\nu} = Z^2\mathcal{R}\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \qquad n_1,n_2 \in \mathbb{Z}^+$$
and nowadays we know that this is only true for hydrogen-like atoms;$^*$ the energy levels of multi-electron atoms depend on both $n$ and $l$.$^\dagger$
The $n$-dependency was later successfully rationalised by the Bohr model, but saying that "the Rydberg formula only works for hydrogen-like atoms because the Bohr model only works for them" is misleading and misses the point, as:
$^*$ In fact, the energy levels of hydrogen are not only dependent on $n$ (due to various small effects such as – but not limited to – spin-orbit coupling, and hyperfine splitting). Wikipedia has a good overview of the topic here and most QM textbooks have a chapter on the hydrogen atom, where they discuss these perturbations to the Hamiltonian and their effects on the energies. Not surprisingly, the inability to explain this was one of the failures of the Bohr model.
$^\dagger$ Of course, there is also a series of approximations here. The energy levels of multi-electron atoms are only approximately described by sums of orbital energies, so the transition energies are only approximately equal to a difference in energy between two orbitals.
A hydrogen-like atom (or ion) is simply any particle with a nucleus and one electron.
That should be sufficient to answer the question at hand, but I thought I should say a bit more, as some of these answers are potentially confusing.
The historical reason why the Rydberg formula only works for hydrogen-like atoms is because it was originally formulated to explain the spectral lines of hydrogen. It was never intended to explain the spectra of multi-electron atoms.
The physical reason, though, is because the Rydberg formula uses energy levels that only depend on the principal quantum number $n$, which has to be a positive integer:
$$\bar{\nu} = Z^2\mathcal{R}\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \qquad n_1,n_2 \in \mathbb{Z}^+$$
and nowadays we know that this is only true for hydrogen-like atoms;$^*$ the energy levels of multi-electron atoms depend on both $n$ and $l$.$^\dagger$
The $n$-dependency was later successfully rationalised by the Bohr model, but saying that "the Rydberg formula only works for hydrogen-like atoms because the Bohr model only works for them" is misleading and misses the point, as:
$^*$ In fact, the energy levels of hydrogen are not only dependent on $n$ (due to various small effects such as – but not limited to – spin-orbit coupling, and hyperfine splitting). Wikipedia has a good overview of the topic here and most QM textbooks have a chapter on the hydrogen atom, where they discuss these perturbations to the Hamiltonian and their effects on the energies. Not surprisingly, the inability to explain this was one of the failures of the Bohr model.
$^\dagger$ Of course, there is also a series of approximations here. The energy levels of multi-electron atoms are only approximately described by sums of orbital energies, so the transition energies are only approximately equal to a difference in energy between two orbitals.
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I think one key aspect that is overlooked in these replies is the Born-Oppenheimer approximation. The nucleus of any atom can be approximated as a point with very few relativistic correction due to its mass. An electron on the other hand is essentially massless (comparatively speaking. I know it is still a fermion and has mass). Once you put more than one electron in any orbital those relativistic corrections have consequences. So ''hydrogen-like'' means any atom that does not need Pauli or Dirac's insight to explain the deviations in the spectra i.e. the bohr model.
I think one key aspect that is overlooked in these replies is the Born-Oppenheimer approximation. The nucleus of any atom can be approximated as a point with very few relativistic correction due to its mass. An electron on the other hand is essentially massless (comparatively speaking. I know it is still a fermion and has mass). Once you put more than one electron in any orbital those relativistic corrections have consequences. So ''hydrogen-like'' means any atom that does not need Pauli or Dirac's insight to explain the deviations in the spectra i.e. the bohr model.
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