Editing Hydrogen atom (section)

=== Bohr–Sommerfeld Model ===
{{Main|Bohr model}}
In 1913, [[Niels Bohr]] obtained the energy levels and spectral frequencies of the hydrogen atom after making a number of simple assumptions in order to correct the failed classical model. The assumptions included:

# Electrons can only be in certain, discrete circular orbits or ''stationary states'', thereby having a discrete set of possible radii and energies.
# Electrons do not emit radiation while in one of these stationary states.
# An electron can gain or lose energy by jumping from one discrete orbit to another.

Bohr supposed that the electron's angular momentum is quantized with possible values:
<math display="block">L = n \hbar</math>  where <math>n = 1,2,3,\ldots</math>
and <math>\hbar</math> is [[Planck constant]] over <math>2 \pi</math>. He also supposed that the [[centripetal force]] which keeps the electron in its orbit is provided by the [[Coulomb's law|Coulomb force]], and that energy is conserved. Bohr derived the energy of each orbit of the hydrogen atom to be:<ref>{{Cite web|url = http://alpha.chem.umb.edu/chemistry/ch115/carter/files/103more/BohrEquations.pdf| title = Derivation of Bohr's Equations for the One-electron Atom|publisher = University of Massachusetts Boston}}</ref>
<math display="block">E_n = - \frac{ m_e e^4}{2 ( 4 \pi \varepsilon_0)^2 \hbar^2 } \frac{1}{n^2}, </math>
where <math>m_e </math> is the [[electron mass]], <math>e </math> is the [[electron charge]], <math>\varepsilon_0 </math> is the [[vacuum permittivity]], and <math>n   </math> is the [[quantum number]] (now known as the [[principal quantum number]]). Bohr's predictions matched experiments measuring the [[hydrogen spectral series]] to the first order, giving more confidence to a theory that used quantized values.

For <math>n=1</math>, the value<ref name="codata">Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor (2019), "The 2018 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 8.0). Database developed by J. Baker, M. Douma, and [[Svetlana Kotochigova|S. Kotochigova]]. Available at http://physics.nist.gov/constants, National Institute of Standards and Technology, Gaithersburg, MD 20899. [http://physics.nist.gov/cgi-bin/cuu/Value?ryd Link to R<sub>∞</sub>], [http://physics.nist.gov/cgi-bin/cuu/Value?rydhcev Link to hcR<sub>∞</sub>]</ref>
<math display="block">\frac{ m_e e^4}{2 ( 4 \pi \varepsilon_0)^2 \hbar^2 } =\frac{m_{\text{e}} e^4}{8 h^2 \varepsilon_0^2} = 1 \,\text{Ry} = 13.605\;693\;122\;994(26) \,\text{eV} </math>
is called the Rydberg unit of energy. It is related to the [[Rydberg constant]] <math>R_\infty</math> of [[atomic physics]] by <math>1 \,\text{Ry} \equiv h c R_\infty.</math>

The exact value of the Rydberg constant assumes that the nucleus is infinitely massive with respect to the electron. For hydrogen-1, hydrogen-2 ([[deuterium]]), and hydrogen-3 ([[tritium]]) which have finite mass, the constant must be slightly modified to use the [[reduced mass]] of the system, rather than simply the mass of the electron. This includes the kinetic energy of the nucleus in the problem, because the total (electron plus nuclear) kinetic energy is equivalent to the kinetic energy of the reduced mass moving with a velocity equal to the electron velocity relative to the nucleus. However, since the nucleus is much heavier than the electron, the electron mass and reduced mass are nearly the same. The Rydberg constant ''R<sub>M</sub>'' for a hydrogen atom (one electron), ''R'' is given by
<math display="block">R_M = \frac{R_\infty}{1+m_{\text{e}}/M},</math>
where <math>M</math> is the mass of the atomic nucleus.  For hydrogen-1, the quantity <math>m_{\text{e}}/M,</math> is about 1/1836 (i.e. the electron-to-proton mass ratio). For deuterium and tritium, the ratios are about 1/3670 and 1/5497 respectively. These figures, when added to 1 in the denominator, represent very small corrections in the value of ''R'', and thus only small corrections to all energy levels in corresponding hydrogen isotopes.

There were still problems with Bohr's model:

# it failed to predict other spectral details such as [[fine structure]] and [[hyperfine structure]]
# it could only predict energy levels with any accuracy for single–electron atoms (hydrogen-like atoms)
# the predicted values were only correct to <math>\alpha^2 \approx 10^{-5}</math>, where <math>\alpha</math> is the [[fine-structure constant]].

Most of these shortcomings were resolved by [[Arnold Sommerfeld|Arnold Sommerfeld's]] modification of the Bohr model. Sommerfeld introduced two additional degrees of freedom, allowing an electron to move on an elliptical orbit characterized by its [[Orbital eccentricity|eccentricity]] and [[declination]] with respect to a chosen axis. This introduced two additional quantum numbers, which correspond to the orbital [[angular momentum]] and its projection on the chosen axis. Thus the correct multiplicity of states (except for the factor 2 accounting for the yet unknown electron spin) was found. Further, by applying [[special relativity]] to the elliptic orbits, Sommerfeld succeeded in deriving the correct expression for the fine structure of hydrogen spectra (which happens to be exactly the same as in the most elaborate Dirac theory). However, some observed phenomena, such as the anomalous [[Zeeman effect]], remained unexplained. These issues were resolved with the full development of quantum mechanics and the [[Dirac equation]]. It is often alleged that the [[Schrödinger equation]] is superior to the Bohr–Sommerfeld theory in describing hydrogen atom. This is not the case, as most of the results of both approaches coincide or are very close (a remarkable exception is the problem of hydrogen atom in crossed electric and magnetic fields, which cannot be self-consistently solved in the framework of the Bohr–Sommerfeld theory), and in both theories the main shortcomings result from the absence of the electron spin.  It was the complete failure of the Bohr–Sommerfeld theory to explain many-electron systems (such as helium atom or hydrogen molecule) which demonstrated its inadequacy in describing quantum phenomena.