3.1.2 The Act Of Desperation Pays Off: Quantum Objects Show Surprising Power To Explain

Had Planck’s explanation of the black body radiation done nothing more than reproduce the Rubens and Kurlbaum data, it may have been quickly forgotten despite its simplicity and its ability to relate that data to well known universal constants such as the electric charge, the Boltzmann's constant, and the speed of light. What catapulted the theory to inevitability was its surprising power to explain everything else which troubled the corpuscular conception at the time.

In a paper published in 1905, Albert Einstein suggested that, if the resonator energy is quantized, so must be the radiation field which stands in equilibrium with a collection of them. In it, he compares the thermodynamic properties of heat radiation of density sufficiently low as to reduce Planck’s formula to that of Wien, with that of an ideal gas:

From this we further conclude that monochromatic radiation of sufficiently low density behaves thermodynamically as if it consisted of mutually independent energy quanta of magnitude $\frac{h c}{λ}$ .
Albert Einstein, On an Heuristic Point of View Concerning
the Production and Transformation of Light (1905)

But, if that were true, the photoelectric effect according to which light shining upon a metallic body extracts electrons from it, could be simply described as follows:

The body’s surface layer is penetrated by quanta whose energy is converted at least partially into kinetic energy of the electrons. The simplest conception is that light quantum transfers its entire energy to a single electron; we will assume that this can occur. If each energy quantum of the incident light transmits its energy to electrons independently of all others, then the velocity distribution of the electrons, that is, of the cathode rays produced, will be independent of the intensity of the incident light; on the other hand, the number of electrons leaving the body will be proportional to the intensity of the incident light.

On this theory, the experimental results which so befuddled classical electromagnetism are therefore precisely the ones which should have been expected: light intensity, which measures the number of energy quanta passing through the unit surface per unit time, will determine the number of electrons it extracts from the metal rather than their energy, while the kinetic energy of each electron, which is the energy in excess of that required to extract the electron from the metal, is directly proportional to the frequency of the light employed.

By 1913, this ability of the new conception introduced by Planck to explain the photoelectric effect was extended by Niels Bohr to the atomic spectra which the classical theory so dismally failed to describe. If Planck’s explanation of heat radiation involved an act of desperation, and Einstein’s explanation of the photoelectric effect was the result of a rational exploration of the consequences of Planck’s formula, Bohr’s explanation of atomic spectra was positively inspired guess work:

That an insecure and contradictory foundation was sufficient to enable a man of Bohr’s unique instinct and sensitivity to discover the principal laws of the spectral lines and of the electron shells of the atoms, appeared to me as a miracle—and appears to me a miracle even today. This is the highest form of musicality in the sphere of thought.
Albert Einstein, Autobiographical Notes, (1949)

Bohr takes for granted two things: the atomic structure suggested by Rutherford's scattering experiments with $α$ particles and the manifest stability of that structure. Thus, he assumes that electrons gravitate around the positive nucleus of the atom under the influence of the Coulomb attraction between them and that, while doing so, the accelerating electrons do not emit any electromagnetic energy. This latter assumption is clearly in direct contradiction with Maxwell’s theory according to which the accelerating electrons should continuously lose energy through emission of electromagnetic waves. By nevertheless making it, Bohr is thus preserving the atomic structure against the eventual collapse of its electrons onto the nucleus. In addition to these two assumptions he makes one more: that only those orbits are allowed for which the electron angular momentum is an integer multiple of Planck’s constant:

m v r = \frac{n h}{2 π} n = 1,2, 3, \dots

Consider the hydrogen atom which has one electron rotating around a nucleus. Bohr assumes the orbit to be circular. Therefore, the Coulomb force will balance against the centrifugal force at all points along the orbit and:

\frac{e^{2}}{r^{2}} = m \frac{v^{2}}{r}

where $e$ is the electric charge of the electron, $m$ is its mass, and $r$ is the radius of the circular orbit. Solving these two equation for the speed $v$ and radius $r$ provides:

r = \frac{n^{2} h^{2}}{4 π^{2} m e^{2}}

v = \frac{2 π e^{2}}{n h}

which allows the evaluation of the total electron energy for each of the allowed orbits as:

E (n) = - \frac{e^{2}}{2 r} = - \frac{2 π^{2} m e^{4}}{n^{2} h^{2}}

Bohr calls these orbits stationary states of the atom because, while in them, the atom emits no radiation.

That leaves open the question of how then does the atomic spectrum ever get generated. Bohr’s answer is that energy is emitted in the form of a quantum of radiation whenever the atom transitions between two stationary states and that the wavelength of that quantum of radiation is related to the difference between the energies of the two stationary states simply as:

E (m) - E (n) = \frac{h c}{λ}

This leads to the following expression for the emitted wavelength:

\frac{1}{λ} = R (\frac{1}{m^{2}} - \frac{1}{n^{2}})

with:

R = \frac{2 π^{2} m e^{4}}{c h^{3}}

This is not only the Rydberg rule for the spectral lines of the hydrogen atom which went unexplained for so long, but also correctly relates the Rydberg constant to the fundamental properties of the electron.

This imaginative explanation is simple, brilliant, and completely unintelligible from the point of view of the then existing corpuscular conception of matter. While Newtonian mechanics does govern the motion of the electron in orbit, those orbits are not free to have any radius one wishes; only stationary states are allowed, a manifestly non-classical idea. This quantization process leads to a corresponding quantization of electron energy in an atom, no less a revolutionary idea than Planck’s quantization of electron energy in a resonator. But perhaps the most intriguing aspect of Bohr’s theory of atomic spectra is the discontinuous jumps that must occur between stationary states. Neither the time when they occur, nor the direction in which the resulting quantum is consequently ejected, could be specified. Despite its simplicity, the Bohr model is not about any particle we had ever encountered before and with which we could thereby had ever become familiar; it is about something totally unfamiliar to us. All these constituents of matter, electrons, photons, and atoms are exceedingly strange. Their introduction into the scientific project is a high price to pay for their explanatory power but, in view of the desperate situation prevailing at the turn of the century, a price the community was quite prepared to pay.