The discovery of the wave nature of particles led Erwin Schrödinger to formulate his equation of the wavefunction in 1926, part of his quest to find a more reliable description of electrons in the atom. This article will lay out the mathematical basis of the wavefunction and its implications for quantum mechanics.
A bead on a string that is fixed at both ends is an example of an infinite potential well. It may move along one direction (call it x) between two limits (call these 0 and L). While a bead has no natural preference for one point on the string over any other, the following derivation will show that electrons do. In a one-dimensional well, their positions fixed between two limits, electrons ‘prefer’ to be in the middle and never occur at either edge. Let’s begin.
The general wavefunction Ψ (pronounced ‘sigh’) of an electron in a one-dimensional well is defined as:A is the maximum height of the wave, k the wavenumber which relates to its wavelength, and x ranges from 0 to L as with the bead on a string.
At this point it’s worth clarifying what the wavefunction is. Although it appears to be fundamental, its physical meaning is unknown, and is the subject of various interpretations which are beyond the remit of this article. Its square, however, is simple and measurable. For a well with one electron, the integral of Ψ² yields the probability of finding the electron in a particular region of the well.
Probability is a familiar concept. In the case of a bead on the string, the probability of finding the bead off the string is zero and thus Ψ² = 0 at these points. Waves, unlike particles, are continuous in space, meaning they have no sudden breaks, and so if Ψ² = 0 just above L and just below 0 then the same must true at these points. So:In case it isn’t apparent, this has shown why particles, whose positional probabilities relate to wave functions, can never appear at the edges of infinite wells. This has the consequence of allowing for k to be defined in simpler terms. If:Then, since sin(x) only equals zero at points where x = n𝜋, then:Thus k can be replaced to give:At this point, it’s worth reflecting on n. n has integer values 1, 2, 3, 4… and its introduction is a consequence of treating the electron probability function as a wave, and by restricting both ends to zero. Since n relates to energy, which is shown later, this is the reason electrons are restricted to precise energy levels in atoms.
Finding A requires only the deduction that the probability of finding the particle in the entire well must be 1, so:The above can be rewritten as:This follows a prolonged derivation, available here. Thus the full wavefunction is:
For n = 1 and L = 1, the probability P of being in a region from x₁ to x₂ is given as:Below this wavefunction is graphed and regional probabilities are given.
~40% of the time the electron will be found in the middle fifth of the well, while ~10% of the time will it appear in the two fifths at both edges. Rather profoundly, the bead on a string is no different. It too obeys Schrödinger’s equation and has its own wavefunction just like the one above. The difference has to do with the associated energies involved, which as mentioned relate to n. Cranking up n yields the classical picture where there is no positional preference, as with beads on strings.In the n = 5 graph, there is equal probability of finding the electron in either fifth, while for the n = 20 graph, the classical picture is re-emerging where the electron can be virtually anywhere. As n is increased into the thousands and millions, the graphs are filled with yellow; the particle can be found anywhere.