# How is Bohr's model related to electron cloud models via the correspondence principle?

I came across a discussion of atomic Rydberg states, which appear to be defined as having their outer electron in a highly excited state. In the same text, it mentioned that these excited electrons could be modeled rather well by the Bohr model.

I don't see why this should be the case. The Bohr model treats the electron as a circularly orbiting body around the nucleus, whereas the electron cloud model seems to produce orbits which are highly anisotropic. In my understanding, the Bohr model also proposes electron orbits with fixed radii.

I don't see how this can be deduced from taking the limit of large $n$. How is Bohr's model related to electron cloud models via the correspondence principle?

On a further side note, I am curious why the different orbital energy splitting as a result of atomic effects (Fine structure, Hyperfine etc) which causes non-degenerate orbitals wrt $\ell$ is ignored for high $n$, where orbital energies are only dependent on the principal quantum number.

References:

1. Randall G. Hulet and Daniel Kleppner, Rydberg Atoms in "Circular" States, Phys. Rev. Lett. 51 (1983) 1430 https://doi.org/10.1103/PhysRevLett.51.1430

2. R.J. Brecha, G. Raithel, C. Wagner, H. Walther, Circular Rydberg states with very large n, https://doi.org/10.1016/0030-4018(93)90392-I

• @CountTo10 Here are a couple of texts which discuss it briefly in the introduction: journals.aps.org/prl/pdf/10.1103/PhysRevLett.51.1430a (Rytlberg Atoms in "Circular" States ,Randall G. Hulet and Daniel Kleppne) and sciencedirect.com/science/article/pii/003040189390392I (Circular Rydberg states with very large n, Brecha et al) Oct 24, 2016 at 19:27
• my answer here is relevant physics.stackexchange.com/q/386927 May 11, 2018 at 7:41

Define $$n_r~:=~n-\ell-1~\geq 0,$$ where $n$ and $\ell$ is the principal and azimuthal quantum number, respectively. Bohr's model works best in the limit

$$\ell \gg 1$$

(to get to the semiclassical limit & the correspondence principle), and

$$n_r \text{ small}$$

(to ensure that the orbital has a well-defined radius).

Could you please accept that I have written this post as an attempt as an answer, basically in order to learn more about the history of quantum models and the connection between Bohr / Rydberg models. Hopefully it will spur someone else to provide a more sophisticated answer that we both can learn from.

I came across a discussion of atomic Rydberg states, which appear to be defined as having their outer electron in a highly excited state. In the same text, it mentioned that these excited electrons could be modelled rather well by the Bohr model.

An explanation of Rydberg and Bohr Atoms and their similiarities

A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number. These atoms have a number of peculiar properties including an exaggerated response to electric and magnetic fields, long decay periods and electron wavefunctions that approximate, under some conditions, classical orbits of electrons about the nuclei. The core electrons shield the outer electron from the electric field of the nucleus such that, from a distance, the electric potential looks identical to that experienced by the electron in a hydrogen atom.

In spite of its shortcomings, the Bohr model of the atom is useful in explaining these properties. Classically, an electron in a circular orbit of radius r, about a hydrogen nucleus of charge +e, obeys Newton's second law:

$${\displaystyle \mathbf {F} =m\mathbf {a} \Rightarrow {ke^{2} \over r^{2}}={mv^{2} \over r}}$$

where $$k = 1/(4πε0)$$.

Orbital momentum is quantized in units of $$ħ$$:

$${\displaystyle mvr=n\hbar }$$.

Combining these two equations leads to Bohr's expression for the orbital radius in terms of the principal quantum number, $$n$$:

$${\displaystyle r={n^{2}\hbar ^{2} \over ke^{2}m}.}$$

It is now apparent why Rydberg atoms have such peculiar properties: the radius of the orbit scales as $$n2$$ (the $$n = 137$$ state of hydrogen has an atomic radius ~1 µm) and the geometric cross-section as $$n4$$. Thus Rydberg atoms are extremely large with loosely boundvalence electrons, easily perturbed or ionized by collisions or external fields.

From the abstract of Circular Rydberg States, which you listed as a text containing the Bohr reference.

Reported is the production of a continuous beam of circular state rubidium Rydberg atoms of principal quantum numbers $$n$$ around $$n=67$$. The circular states are populated using crossed electric and magnetic fields. They are detected continuously by a novel field ionization scheme. The circular character of the atoms is derived from the field ionization patterns, and from microwave spectra of the transitions to circular states with lower n. The circular Rydberg atoms with very large n shall be used for studies of microwave ionization and for one-atom maser experiments.

From this, it seems they created circular states, a la the Bohr model, using techniques that, by co-incidence (or the properties of their equipment) mimicking the circular orbits, and distorting the proper electron cloud model. The Wikipedia articles mentions Rydberg atoms susceptibility to this.

• Thanks for the post. Taking a further look at a couple of books/notes, I believe that the notion of a circular orbit originates from the requirement that $l$ must also be large and of the order of $n$ (often the requirement $l=n-1$ is mentioned alongside $n$ large). On pg 5 of notes (following comment) it seems that the predicted Bohr radius is obtained from the Hydrogen orbital functions for $<r>$ if $n$ and $l$ are large, one example of the correspondence principle. However, this does not help understand why we can assume the orbits of these Rydberg states are approx. circular. Oct 24, 2016 at 21:34
• Relevant notes : phys.spbu.ru/content/File/Library/studentlectures/schlippe/… Oct 24, 2016 at 21:34