Deriving c

Screen Shot 2017-06-14 at 13.50.39Physics has its occasional eureka moment. James Clerk Maxwell had one such moment in 1862 when he discovered that the equations he had found describing electric and magnetic waves combined to describe light. The speed of light c, this article will demonstrate, can easily be found from Maxwell’s equations. As this derivation is an undergraduate degree exam question each step will be described qualitatively, but a brief glance at the vector operators guide in Resources is recommended.

One of the most profound 19th century discoveries was Faraday’s finding that electric currents passing through wires induce magnetic field loops, as shown below.Screen Shot 2017-06-14 at 15.16.15By 1862 Maxwell had shown that the gradient of the magnetic field H was directly proportional to the curl (or rotationality) of the electric field E. This essentially means that electricity and magnetism are one and the same, and a magnetic field flowing around a wire will equally induce a current through it, as described by:
Screen Shot 2017-06-14 at 15.20.08Where μ0 is the magnetic permeability constant, defined later. Taking the curl of both sides gives:Screen Shot 2017-06-14 at 15.20.14The curl of H (∇ x H) is defined by Ampére’s law as:Screen Shot 2017-06-14 at 15.29.30Where σ is the conductivity of the medium and ε₀ is the electric permittivity constant. Inserting this expression above gives:
Screen Shot 2017-06-14 at 15.29.36Using the following identity the Laplacian can be extracted. However, imposing the vacuum condition means that σ = 0 and the divergence of E (∇ . E) = 0.Screen Shot 2017-06-14 at 15.31.03Thus:Screen Shot 2017-06-14 at 15.31.11What has been derived is a wave equation, describing an electromagnetic wave, where μ0 = 1.26 x 10-6 N A⁻² and ε0 = 8.85 x10-12 F m⁻¹. Notice the similarity to the general equation for a wave:Screen Shot 2017-06-14 at 15.33.02In this equation v is the speed of the wave. The speed of an electromagnetic wave can therefore be found:Screen Shot 2017-06-14 at 15.33.41Screen Shot 2017-06-14 at 15.33.52This gives v = ~ 3 x10⁸ m s⁻¹, the speed of light. Thus, light can be treated as an electromagnetic wave. The century and a half since Maxwell first worked out this derivation has corroborated and built upon it.

Infinite Wells

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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:Screen Shot 2017-06-09 at 14.58.54A 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.
Screen Shot 2017-06-09 at 15.07.33At 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.
Screen Shot 2017-06-12 at 15.36.07Probability 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:Screen Shot 2017-06-12 at 15.36.14In 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:Screen Shot 2017-06-12 at 15.50.05Then, since sin(x) only equals zero at points where x = n𝜋, then:Screen Shot 2017-06-12 at 15.53.12Thus k can be replaced to give:Screen Shot 2017-06-12 at 15.58.23At 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:Screen Shot 2017-06-12 at 16.17.22The above can be rewritten as:Screen Shot 2017-06-12 at 16.19.48Screen Shot 2017-06-12 at 16.47.41This follows a prolonged derivation, available here. Thus the full wavefunction is:
Screen Shot 2017-06-12 at 16.49.48For n = 1 and L = 1, the probability P of being in a region from x₁ to x₂ is given as:Screen Shot 2017-06-12 at 16.53.04Screen Shot 2017-06-12 at 16.55.41Below this wavefunction is graphed and regional probabilities are given.
figure_1~40% 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.figure₂figure_3In 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.

The equation relating energy, mass and n is derived here, and a basic infinite well simulator, designed to produce the above graphs, can be found in Resources.