Key Points from this Chapter
Based on analogy with the momentum of a photon, Louis de Broglie proposed that particles can be described by “matter waves” with an associated wavelength (λ), linked to the momentum of the particle (p) by:
λ = h / p
(where p is Planck’s constant)
The wave character of electrons, and the validity of the de Broglie equation to describe their wavelength, are supported by diffraction of an electron beam by a crystal.
A particle can therefore be described by a wave function (Ψ) which is a function of space and time. In a stationary situation, such as when an electron is localised in an atom, the space and time parts of the function can be separated: Ψ =ψ(x).f(t). The time part, in this case, is a simple oscillation, so we can focus on the space part, which we call the waveform.
The wave function can be interpreted (according to the Born interpretation) in terms of the probability that a particle will, if we use some means to pin it down, be detected at a particular point in space: ψ2(x) = probability density that particle would be located at point x.
A simple sinusoidal wave extends indefinitely but a spatially localised waveform – a wave packet - can be created by superposing waveforms of different wavelength. However it follows from the de Broglie equations that this introduces an uncertainty in the momentum associated with the waveform.
Narrowing the spatial extent of a wave packet requires increasing the range of wavelengths of the superposed waves and therefore increasing the uncertainty in the momentum. This is quantified in the uncertainty principle, which relates the standard deviations in the position of a particle and the momentum associated with the superposed waves:
σxσp ≥ h / 4π
It follows from the uncertainty principle that narrowing the spatial focus of a particle requires an increase in the mean momentum associated with the superposed waves that contribute to its wavefunction. Since momentum is related to kinetic energy (Ek = p2/2m), it follows that localising particles comes at an energy cost (the localisation energy). The narrower the region of space in which a particle is confined, the higher the kinetic energy of its wavefunction.
This is the reason that an electron atom does not collapse into the nucleus of an atom. The mean distance of an electron from the nucleus is a compromise between electrostatic potential energy and localisation energy.