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Potential Energy Surfaces and Conical Intersections

Quantum Theory

1.1 Quantisation of energy

Quantum theory was initially formulated in 1900 by the German physicist Max Planck in response to the problem of the radiation emitted by a ‘blackbody’ (so called because it absorbs all radiation that falls upon it; it is also a perfect emitter).
Planck proposed that light was concentrated in ‘packets’ of energy, E = hn (n being the frequency of the light). This was in contrast with the previously held belief that light was an electromagnetic wave, arrived at by Thomas Young in 1801, as a result of diffraction and interference experiments. These packets were called quanta, or photons.[1] Planck’s work introduced a new fundamental constant; h, Planck’s constant.
The hypothesis successfully explained the blackbody radiation function, and was further corroborated when Einstein later used the idea in 1905 to explain the photoelectric effect (below),[2] where the kinetic energy of electrons ejected from a metal surface correlates not with the intensity of the light shining on the surface, but with the frequency of the incident light. He received the Nobel Prize in Physics, 1921, for this work.
Planck was awarded the Nobel Prize for Physics, 1918 for his work.

The photoelectric effect
© Copyright Physics 2000, University of Colorado

The combination of the two hypotheses implied that the discrete frequencies found in atomic line spectra implies that the atom, as a quantum body, has certain, discrete ‘energy levels’, and can only absorb or emit energy in certain, discrete amounts, which are multiples of hn.
Niels Bohr used quantum theory in 1913 to explain the hydrogen atom and its atomic spectrum.[3] Rutherford had proposed the classic ‘planetary’ picture of the hydrogen atom (with a positive nucleus and orbiting electron) in 1911, but this was doomed according to classical mechanics – an accelerated charged particle, such as the electron (whose constant angular speed dictates a constantly changing acceleration vector), radiates energy in the form of electromagnetic waves. Hence, the electron should be continually losing energy, and would therefore spiral inwards towards the nucleus as the atom collapses. Bohr assumed that the electron could only move in fixed orbits. This led to the Bohr frequency condition (Equation (1.1)), where the photon of light emitted upon transition from an upper to lower orbit (energy level) has a fixed energy, and therefore also frequency and wavelength.

Equation 1.1 (1.1)

Bohr was also awarded the Nobel Prize in Physics, 1922, for this work.

1.2 Special relativity and wave-particle duality

Later, in 1923, de Broglie suggested that electrons (and photons, themselves also being particles) might have a wave property associated with them, as a result of applying Einstein’s special relativity, E = mc2.[4]
The de Broglie wavelength associated with a particle of mass m and speed n is given by equation (1.2):

Equation 1.2 (1.2)

…where p is the linear momentum. The equation also holds for a photon travelling at the speed of light, c. The photon differs from the electron in that it always travels at speed c and has zero rest mass (but can have appreciable relativistic mass), whereas the electron has a speed less than c and a non-zero rest mass.
This introduced the concept of wave-particle duality; that an electron can behave like both a wave and a particle. This embodies the fact that classical physics cannot provide a picture of quantum particles that we as humans can successfully visualise.
de Broglie’s suggestion was proven by Davisson and Germer,[5] and also by G. P. Thomson, in experiments showing that the interference pattern obtained from diffraction of a beam of electrons from a metal foil was characteristic of the de Broglie wavelength, l = h / p. Davisson and Thomson were jointly awarded the Nobel Prize in Physics, 1937, for this work.

1.3 The uncertainty principle

The uncertainty principle was discovered in 1927 by Werner Heisenberg.[6] It states that, as a consequence of the wave-particle duality of microscopic ‘particles’, there is a limit as to how accurately the position and momentum of such particles can be simultaneously defined. It is typically represented mathematically in the form shown in equation (1.3):

Equation 1.3 (1.3)

The uncertainty principle can be demonstrated experimentally by the attempt to measure simultaneously the x coordinate and the x component of linear momentum, of a microscopic particle, as a beam of such particles is passed through a diffraction grating or slit (see Levine[7]).

1.4 Further information and indeterminacy

The wave nature of a particle can be demonstrated by a simple two-slit diffraction experiment - an explanation of this is given at the Physics Department at Trinity College, Dublin's mini-site about quantum theory, which you can find here. There are also some great interactive applets at the University of Colorado's “Physics 2000” site (in the Interference Experiments section of the Atomic Lab) which illustrate these effects. The 'electron gun' applet, illustrating two-slit diffraction of particles is linked here for convenience:
(If you have trouble running this Java applet, check the System Requirements & Troubleshooting page at Physics 2000).

Turn on the electron gun and adjust the gap between the slits using the slider.
Use the '+' and '-' keys on your keyboard to control gun rate.
Press the "backspace" key to wipe the phosphor screen.
© Copyright Physics 2000, University of Colorado

For a given spot on the phosphor screen, we cannot know which slit the wave/particle that caused it has come from. This is indeterminacy, and is a fundamental consideration in the difference between macroscopic and quantum ideas. It is often conceptualised using the famous Schrödinger's cat ‘thought experiment’.

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[1]  M. Planck, “Über das Gesetz der Energieverteilung im Normalspektrum”, Ann. d. Physik, 4, 553, 1901
[2]  A. Einstein, “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt”, Ann. d. Physik, 17, 132, 1905
[3]  N. Bohr, Fysisk Tidsskrift, 12, 97, 1914
[4] L. de Broglie, Comptes rendus de l'Académie des Sciences, 177, 507-510, 1923
[5] HyperPhysics, Wave nature of the electron: http://hyperphysics.phy-astr.gsu.edu/hbase/davger.html#c1
[6] W. Heisenberg, Z. Physik, 45, 172, 1927
[7] Ira N. Levine, “Quantum Chemistry”, 5th edition, Prentice-Hall International (UK) Limited, 2000