Classical Quantum Theory

I think I can safely say that nobody understands quantum mechanics.

--R. Feynman

Though it had humble beginnings, quantum theory rose to challenge and topple the most sacred opinions of physics. Quantum theory has had profound effects on our society as well, from the dilemmas it poses philosophers to the breakthroughs that allow one to see atoms. Applications of quantum theory promise computing power millions of times what we have currently as well as help us to determine what the universe is and where we came from.

A quantum is a discrete number or state. This is in opposition to the idea of a continuum which is the collection of an infinite number of states. Whereas anything is possible in a continuum, it is only possible to have an integer multiple of the quantum.

In 1811, Amedeo Avogadro proposed that substances were made up of trillions of trillions of atoms held together. (6.02 e 23 particles per mole), thus describing the "quantization" of matter. In 1897, J.J. Thompson demonstrated the charge to mass ratio of an electron, implying the "quantization" of electric charge. (In 1909, R. A. Millikan demonstrated the quantization of electric charge with his immortal oil drop experiment.) However, neither of these results had a great effect on the scientific community, because they could both be explained by the classical laws of mechanics, thermodynamics, and electricity.

The Ultraviolet Catastrophe

At the end of the 19th century, a purely classical and consistent law, termed the Rayleigh-Jeans law was developed to describe the emission of radiation from an opaque body. The Rayleigh-Jeans law gave splendid results for very high frequency radiation, but the results were nowhere near the empirical data for lower wavelengths. This was called the ultraviolet catastrophe, and it was the first sign that the classical theory might not be completely correct.

In 1900 the German physicist Max Planck announced that by making some odd assumptions, he was able to make the theory fit with experiment. His assumption was that the energy of the radiation was quantized, that is the energy was a discrete variable, given by

En = nhf, where n =0, 1, 2, ..., f is the frequency of the light,
and h is Plank's Constant, 6.626 e -34 J-sec

Planck didn't believe his own results and tried in vain to reconcile his theory with classical physics. He spent much of his later life focused on religion, trying to put his past behind him.

The Photoelectric Effect

Another unexplained result came about at the end of the 19th century. It was discovered that if certain types of light were incident on certain metals, cathode rays (electrons) were produced. The odd thing was that the production of these rays depended not on the energy of the light, but only upon its frequency.

In 1905, Albert Einstein explained the photoelectric effect making use of the assumption of the quantization of radiation proposed by Planck. He called each quanta of light having energy hf a photon. He stated that when a photon penetrates the metal, all of its energy is given to an electron. If this energy is greater than the amount of energy needed to remove the electron from the surface (called F, the work function, which is typically on the order of a few electron volts for most metals), the electron will be emitted with an energy given by:

mv2/2 = hf -F

This theory explained why even very intense light sources at low frequencies couldn't eject electrons, while extremely weak light sources at higher frequencies could. (Einstein won his Nobel Prize in physics for this theory.)

Quantum Models of the Atom

If you observe the radiation emitted by atoms of a gas excited by an electrical discharge, or by atoms in a flame, by using a spectroscope, you will notice that certain wavelengths of light are seen, not a continuous spectrum. In 1885 a Swiss schoolteacher, Johann Balmer, developed an empirical formula to describe this for hydrogen. Later, a more general expression was found.

Many attempts were made to come up with a theoretical description of the process by which this occurred. Electromagnetic theory showed that charges radiate when accelerated, so J.J. Thomson developed a model in which electrons were embedded in a kind of fluid atom which contained most of the mass of an atom and enough charge to balance out the electrons.

Thomson could never get his theory to accurately explain these results, and it was ruled out unquestionably after Ernest Rutherford's experiment of 1913. This famous experiment fired alpha particles at a sheet of gold foil, and it showed that the atom contained a very small, very dense nucleus, surrounded by electrons.

This posed a great problem to the theory of the atom. If an electron is moving around the outside of a nucleus and is radiating energy, its orbit will decay until the electrons crashes into the nucleus.

Later in 1913, Danish physicist Neils Bohr developed a theory which explained the behavior. He postulated that the electron could move in certain orbits without radiating. He then assumed that the atom radiates when the electrons somehow make a transition from one orbit to another. Further he postulated that the frequency of radiation is related to the energies of the orbits by:

hf = Ei - Ef

Bohr further developed tools to measure the energy of the orbit and the radius of orbit:

E = -kZe2/2r, En = -Z2Eo/n2

where Eo is the energy of the ground state of hydrogen, 13.6 eV (ionization energy)

r = n2ao/Z

where ao =0.529 A, the Bohr radius. The integer, n, is called the fundamental quantum number.

These results completely explained the earlier quantum spectra, and made quantum theory an integral part of the description of the atom.

Overview of Classical Quantum Theory

Insofar as it went, classical quantum theory did a good job of providing a way to explain and predict the results of experimental data. However, classical quantum theory was really little more than a mixture of classical theory and quantum assumptions; no true paradigm shift had occurred. That is, people were not thinking about the world in a different manner, but were trying to modify their current beliefs to match the new data. The events of subsequent years in the field would make that effort impossible ...