

c = speed of light m = mass E = energy h = Planck’s constant p = linear momentum of an electron Q = density of wave 
ν = light frequency of a bound electron in Hertz, l/c nu ω = angular frequency (2πν), l/c omega π = pi λ, l/c lambda, for wavelength Ψ for wave amplitude, the Schrodinger wave function, u/c psi 


Quantum mechanics from its very beginning differed from classical physics in that it did not start from the treatment of individual (micro) objects, and then applied statistical considerations to those laws. It gave, in general, only statistical predictions without reference to the underlying laws. The following is based upon David Bohm’s Quantum Theory (Dover Publications, 1989).
The first evidence in favour of quantum theory came around 1900 from the work of Max Planck who came to the conclusion that light transmits energy not continuously but in “quanta” or bundles of size E = hν. In 1905 Einstein published a paper that accepted the full consequences of Planck’s hypothesis by stating that radiation acts as if it consists of quanta of energy E = hν, which occur only in emission and absorption, but can have an independent existence in empty space.
Evidence of the interference and diffraction of light waves had already been derived by an experiment in which light was shone directly on two slits. If the first slit was open there is more or less uniform light variation, but if both slits were open there were sets of alternating light and dark areas.
Further conflicting results were obtained in the following experiment. Light was shone on a metal surface A, placed in a glass tube evacuated of air. Another plate B was placed in the tube to collect any electrons liberated from the illuminated surface A. When the plate was illuminated with light of frequency ν, the electrons all gained the same amount of energy E = hν, which depended only on the frequency of the light, not on its intensity. Thus, when the light was very weak, the electrons still gained the same energy E = hν, but correspondingly fewer electrons were liberated.
This result is in direct contradiction of the preconditions of classical theory, which state that the energy gained should be proportional to the intensity of radiation. With weaker light less energy should be gained. This experiment suggests that light does not consist of waves but swarms of particles.
The next step was by the Danish scientist, Niels Bohr. Investigations had already led to the conclusion that matter is made of atoms, and that these atoms in turn include small negatively charged particles called electrons, which circulate around a heavier positively charged nucleus in much the same way as planets orbit the sun. As the electron goes around the nucleus it should emit electromagnetic waves at the same frequency as the rotation. The frequency was calculated as being around 10^{16} cycles per second, the same order of magnitude as the frequency of light.
When the process of light emission was studied in more detail, a number of serious contradictions between existing theory and experimental results became apparent. According to classical theory, a moving charged particle, such as an electron, should continuously lose energy by radiating electromagnetic waves.
Another important contradiction between classical theory and experimental results arose in a detailed study of the frequencies of radiation emitted by atoms. According to the orthodox theory there should be a continuous range of possible sizes of orbits of the electrons, and a correspondingly continuous range of light frequencies. In reality only a finite number of discrete frequencies are emitted. Bohr reasoned that only certain size orbits were possible: he postulated a smallest possible orbit, which he denoted by a_{0}, with the lowest possible energy. Once the electron entered this orbit it would not be able to lose any more energy and it would remain in this orbit until it was disturbed from outside. For example, if an electron was in orbit a_{2} with energy Ea_{2}, and then moved to a smaller, intermediate orbit a_{1}, it would radiate the full energy difference E(a_{2} − a_{1}) in one quantum of light, with the frequency of the emitted light being given by the Planck relation E(a_{2} − a_{1}) = hν.
Physicists were forced to take Bohr’s model seriously when it was discovered that a characteristic spectrum of light is obtained from any element in gaseous form when an electric charge is passed through it. Each element has its own unique series of spectral lines: this, indeed, is the basis of modern chromatography.
A step towards the better understanding of discrete energies of atomic orbits was made by Louis de Broglie, who suggested that just as lightwaves had a particlelike character, atomic particles might also have a wavelike character. He was guided by the appearance of sets of discrete frequencies in many kinds of classical waves, for example, a fixed string must vibrate in integral multiples or harmonics of a fundamental frequency. In 1897 J. J. Thompson proposed that the electron was a particle and his son G. P. Thompson, thirty years later, postulated that the electron was a wave.
De Broglie proposed the existence of a new kind of wave connected with the electron. If this wave is confined within the atom, it will have discrete frequencies of oscillation.
Like Bohr, de Broglie did not rely on advanced mathematical analysis but on innovative ideas about physical reality. He began by accepting two simple expressions for the energy of a particle at rest, e = mc² and e = hν. He accepted Einstein’s special theory of relativity, which requires these expressions to be invariant under a Lorentz transformation.
According to Walter Moore, in Schrodinger: Life and Thought (Cambridge University Press, 1989), since c and h are constants, m and ν must follow the same transformation law. For m_{0} this is
where ß = ν/c. Similarly
The real or apparent slowing up of a moving clock relative to the stationary observer is represented by a wave.
A wave moving in the x direction can be represented by the function sin ω where the phase ω = 2π(νt − x/λ). The change in ω when the particle moves a distance dx = νdt is
dω = 2π(νdt − dx/λ)  =  2π  [  m_{0}c²  dt  −  h  dx  ] 
h  (1 − ß²)^{½}  λ 
De Broglie postulated a theorem of the harmony of phase according to which the phase of the wave associated with the particle must always accord with the phase of a clock at the position of the particle (as measured by an observer at rest). The phase change of the clock in time dt would be
When the two coordinate systems coincide, dω = dω_{1} and dt = dx, and then:
The resulting equality p = h/λ is the famous Broglie equation which relates the momentum of a particle to its wavelength, applying it to electron waves as it does to light waves. Hence discrete frequencies imply discrete energies. This is analogous to the vibration of a string at certain frequencies, and implies resonance within the atom.
Can anyone offer a fuller explanation, or find fault in the mathematics? Comments are invited. Paul Ballard