Editing Bohr model (section)

==Development==

Next, Bohr was told by his friend, Hans Hansen, that the Balmer series is calculated using the Balmer formula, an [[empirical]] equation discovered by [[Johann Balmer]] in 1885 that described wavelengths of some spectral lines of hydrogen.<ref name="aip.org" /><ref name="Rosenfeld1963">{{Cite book |last1=Bohr |first1=Niels |title=On the Constitution of Atoms and Molecules ... Papers of 1913 reprinted from the Philosophical Magazine, with an introduction by L. Rosenfeld |last2=Rosenfeld |first2=Léon Jacques Henri Constant |date=1963 |publisher=Copenhagen; W.A. Benjamin: New York |oclc=557599205}}{{page needed|date=February 2022}}</ref> This was further generalized by [[Johannes Rydberg]] in 1888, resulting in what is now known as the Rydberg formula.
After this, Bohr declared, "everything became clear".<ref name=Rosenfeld1963/>

In 1913 [[Niels Bohr]] put forth three postulates to provide an electron model consistent with Rutherford's nuclear model:

# The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to what [[classical electromagnetism]] suggests. These stable orbits are called stationary orbits and are attained at certain discrete distances from the nucleus. The electron cannot have any other orbit in between the discrete ones.
# The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant: <math> m_\mathrm{e} v r = n \hbar </math>, where <math>n= 1, 2, 3, ...</math> is called the [[principal quantum number]], and <math>\hbar = h/2\pi</math>. The lowest value of <math>n</math> is 1; this gives the smallest possible orbital radius, known as the [[Bohr radius]], of 0.0529&nbsp;nm for hydrogen. Once an electron is in this lowest orbit, it can get no closer to the nucleus. Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper,<ref name="aip.org" /><ref name=Heilbron2013/><ref name="Nicholson1912" /><ref name=McCormmach1966/> Bohr<ref name="bohr1" /> was able to calculate the [[#Electron energy levels|energies of the allowed orbits]] of the hydrogen atom and other [[#Shell model (heavier atoms)|hydrogen-like]] atoms and ions. These orbits are associated with definite energies and are also called energy shells or [[energy level]]s. In these orbits, the electron's acceleration does not result in radiation and energy loss. The Bohr model of an atom was based upon Planck's quantum theory of radiation.
# Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency <math>\nu</math> determined by the energy difference of the levels according to the [[Planck relation]]: <math>\Delta E = E_2-E_1 = h \nu</math>, where <math>h</math> is the Planck constant.

Other points are:
# Like Einstein's theory of the [[photoelectric effect]], Bohr's formula assumes that during a quantum jump a ''discrete'' amount of energy is radiated. However, unlike Einstein, Bohr stuck to the ''classical'' [[Maxwell's equations|Maxwell theory]] of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels; Bohr did not believe in the existence of [[photon]]s.<ref>{{Cite book |last=Stachel |first=John |title=Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle |date=2009 |publisher=Springer |location=Dordrecht |page=79 |chapter=Bohr and the Photon}}</ref><ref>{{Cite book |last=Gilder |first=Louisa |title=The Age of Entanglement |year=2009 |page=55 |chapter=The Arguments 1909—1935 |quote="Well, yes," says Bohr. "But I can hardly imagine it will involve light quanta. Look, even if Einstein had found an unassailable proof of their existence and would want to inform me by telegram, this telegram would only reach me because of the existence and reality of radio waves."}}</ref>
# According to the Maxwell theory the frequency <math>\nu</math> of classical radiation is equal to the rotation frequency <math>\nu</math><sub>rot</sub> of the electron in its orbit, with [[harmonics]] at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels <math>E_n</math> and <math>E_{n-k}</math> when <math>k</math> is much smaller than <math>n</math>. These jumps reproduce the frequency of the <math>k</math>-th harmonic of orbit <math>n</math>. For sufficiently large values of <math>n</math> (so-called [[Rydberg state]]s), the two orbits involved in the emission process have nearly the same rotation frequency, so that the classical orbital frequency is not ambiguous. But for small <math>n</math> (or large <math>k</math>), the radiation frequency has no unambiguous classical interpretation. This marks the birth of the [[correspondence principle]], requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.
# The [[BKS theory|Bohr–Kramers–Slater theory]] (BKS theory) is a failed attempt to extend the Bohr model, which violates the [[conservation of energy]] and [[Conservation of linear momentum|momentum]] in quantum jumps, with the conservation laws only holding on average.

Bohr's condition, that the angular momentum be an integer multiple of <math>\hbar</math>, was later reinterpreted in 1924 by [[de Broglie]] as a [[standing wave]] condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:
: <math>n \lambda = 2 \pi r.</math>

According to de Broglie's hypothesis, matter particles such as the electron behave as [[Matter wave|waves]]. The [[wikiversity:De Broglie wavelength|de Broglie wavelength]] of an electron is
: <math>\lambda = \frac{h}{mv},</math>
which implies that
: <math>\frac{nh}{mv} = 2 \pi r,</math>
or
: <math>\frac{nh}{2 \pi} = mvr,</math>
where <math>mvr</math> is the angular momentum of the orbiting electron. Writing <math>\ell</math> for this angular momentum, the previous equation becomes
: <math>\ell = \frac{nh}{2 \pi},</math>
which is Bohr's second postulate.

Bohr described angular momentum of the electron orbit as <math>2/h</math> while [[Matter wave|de Broglie's wavelength]] of <math>\lambda = h/p</math> described <math>h</math> divided by the electron momentum. In 1913, however, Bohr justified his rule by appealing to the correspondence principle, without providing any sort of wave interpretation. In 1913, the wave behavior of matter particles such as the electron was not suspected.

In 1925, a new kind of mechanics was proposed, quantum mechanics, in which Bohr's model of electrons traveling in quantized orbits was extended into a [[matrix mechanics|more accurate model]] of electron motion. The new theory was proposed by [[Werner Heisenberg]]. [[Schrödinger equation|Another form]] of the same theory, wave mechanics, was discovered by the Austrian physicist [[Erwin Schrödinger]] independently, and by different reasoning. Schrödinger employed de Broglie's matter waves, but sought wave solutions of a three-dimensional wave equation describing electrons that were constrained to move about the nucleus of a [[hydrogen-like atom]], by being trapped by the potential of the positive nuclear charge.