de Broglie's Waves

Around 1900, folk noticed, studied and began making some sense of various processes – archetypically, the character of thermal (black-body) radiation at high frequencies, the photo-electric effect and radioactivity – which involve things whose character mixes up particle-like and wave-like behaviour. These were the two styles of model for which folk had rich toolsets, but it took a while for folk to build an understanding of how to combine the two into a unified style.

In 1904, Planck pointed out that light appeared to be interacting with matter only in ways that transfer an amount of energy equal to a certain constant times the frequency of the light. That constant has come to be known as Planck's constant; its usual denotation is the letter h and its value, measured in Joule seconds, is just under two thirds of ten to the minus 33. In 1927, de Broglie pointed out a relationship, where a process is described partially as a particle and partially as a wave process, between the momentum, p, of the particle involved and the wavelength, L, of the wave process: when you multiply them together, you get Plank's constant: p.L = h. When you take into account the direction of the periodicity described by the wavelength, it turns out to be parallel to p: so p/h has length 1/L and is in the direction of L.

Now it's usual to discuss Planck's constant as having units momentum.length; but the above gives it units of momentum.length/phase. Indeed, equally, Planck's original definition of the constant used it in E = h.f with E an energy and f a frequency; while it is common to treat f as having dimensions 1/time, it's properly got dimensions phase/time; thus h = E/f has dimensions Energy.time/phase, which works out the same as momentum.length/phase. Thus what's commonly given as Planck's constant is just h.cycle; indeed, much of quantum mechanics actually works with Dirac's constant (denoted by ℏ, a crossed h), which is Planck's divided by 2.π, i.e. h.radian with my reading of h's units. Likewise, the orthodox wave vector (see below) emerges as k = g(p)/(h.radian).

To go any further with this, we need to describe wave-like things; and the characteristic thing we're really dealing with is periodic structure, so it's time to introduce the mathematical tool that describes that nicely: wave vectors or, more strictly, covectors.

Periodic processes

When a system is described as having a wavelength, there's an implicit statement that, at least approximately and at least locally, the system repeats itself – and does so in a way characterized by a minimal displacement between points on distinct wave-fronts. This gives us a vector quantity, L: its length is the wave-length and its direction is orthogonal to each wave-front.

The notion orthogonal depends on our having a geometry: when this is described in terms of a metric, g, this latter is a linear isomorphism from vectors to covectors; these last being linear maps from vectors to scalars. This means we can take two vectors (displacements), u and v, and combine them with g to get a scalar (number) g(u,v), which varies linearly with u and linearly with v – so g(2.u,v) = 2.g(u,v) = g(u,2.v), g(u+w,v)= g(u,v)+g(w,v) and g(u,v+w)= g(u,v)+g(u,w). The value of g(u,v) is called the inner product of the two vectors, u and v: and it is usual to work with a symmetric metric, so that g(u,v) = g(v,u). If g(u, v) = 0 then u and v are orthogonal; if g(u, v) is zero for every displacement v within a wave-front, then u is orthogonal to that wave-front.

If we look at an arbitrary displacement in a periodic system, we can ask for the number of wave-fronts it crosses: and we can interpolate between wave-fronts so that the answer is a scalar which is linear in the displacement, with the integers on either side of the scalar being the numbers of wave-fronts that equal displacements can cross when starting in different places. Indeed, this scalar is given by g(L,s)/g(L,L) for an arbitrary displacement s: and the wave process's behaviour is synchronized, between two localities, precisely when the displacement between the two localities has an integer value for this quantity.

Now, formally, g(L,s) is g(L)(s), the result of passing s to the function g(L), which is a covector, i.e. a linear map from vectors (e.g. the displacement, s) to scalars. So we can write the linear map g(L,s)/g(L,L) ←s as K = g(L)/g(L,L), with K(L) = 1. Since g is an isomorphism, it has an inverse: let this be f and you find that f(K)= L/g(L,L), so that K(f(K)) = K(L)/g(L,L) = 1/g(L,L). Thus L= f(K).g(L,L) = f(K)/K(f(K)), so we can obtain L from K: so either of K and L tells us everything the other tells us. Since what really matters is that we can identify displacements between whose ends the wave system is self-synchronized, and K lets us do this without referring to the metric – or, to put it another way, K(s) is nicer to write than g(L,s)/g(L,L) – it makes sense to work in terms of K rather than L.

Not only does this free us from involving the metric – it also frees us from some complications that would arise if the direction of L were light-like in a relativistic context: in that case g(L,L) is zero, as is the wavelength it implies. Indeed, when discussing light (for which this holds) it is usual to discuss the wave-length seen in a space-like cross-section of the system, which depends on the choice of cross-section but does tell us what we want to know, given prior knowledge that L is light-like. The covector K is still defined in this case: indeed, the space-like wavelength is just the minimal displacement, s, in the chosen cross-section for which K(s)=1.

A minor refinement: our wave covector measures the number of cycles of a periodic process, so it actually maps a vector to a scalar-like quantity with units cycle which are essentially similar to units of angle, with the cycle corresponding to the turn. Indeed, properly, the units of wave-length are distance/cycle; for a simple displacement s, g(s,s) has units of length.length, but we should really use a displacement/cycle as L, so g(L,L) is length.length/cycle/cycle and K = g(L)/g(L,L) has units cycle/length; feeding it s as input gives K(s) with units cycle.

It is often practical to model periodic processes in terms of the sinusoid functions, which make it natural to use the radian as unit; 2.π.radian = turn. Consequently it is common to describe periodic processes in terms of a wave covector, k, which maps displacements to the number of radians'-worth of phase-change; if K(s) is 1 cycle, k(s) will be 2.π and dimensionless, so k is K/radian, or at least 2.π.K/cycle. Two locations in the periodic structure will be synchronized precisely if k maps the displacement between them to a multiple of 2.π, or K maps that displacement to a whole number of cycles. While k and K are covectors, it is common to visualize them in terms of the associated vectors f(k) and f(K), with f the inverse of the metric, g: for example, f(K) = L/g(L,L), which is parallel to L but its length is the reciprocal of L's length. However, I prefer to discuss phase as having units; while I shall take it that cycle is synonymous with turn, hence cycle/radian is 2.π, I'll primarily work in terms of K, rather than k.

Restating de Broglie

Strictly speaking, de Broglie only tells us the magnitude of the spatial components of the wave covector, as a function of the magnitude of the spatial momentum. That is actually sufficient to enable us to do some practical physics, such as building a simple model (or two) of the hydrogen atom. However, since both momentum, a vector, and the wave covector have direction as well as magnitude, we should also look at what we can learn about direction.

Using nicely normalized co-ordinates, making g diagonal([−1,1,1,1]), and normalizing displacements to length (rather than time) units, what de Broglie and Planck had observed about a particle can be stated as:

in a single instant of time, a space-like snapshot

has periodic structure whose wave-length times the matter's 3-momentum, p3, is Planck's constant. This gives us a spatial displacement, S, for which K·S is one, g(S,S).g(p3,p3) = h.h and, for any spatial displacement perpendicular to S, i.e. any R with g(R,S) = 0, K·R is zero.

at any fixed position, its time-like story

is periodic; the period times the matter's total energy, E, is h. This gives us a time-wards displacement T with K·T equal to 1 and −g(T/c,T/c).E.E = h.h.

Thus, for a displacement R: if g(S,R) = 0 and g(T,R) is zero, then so is K·R. This covers two degrees of freedom; and the specifications of T and S cover the rest. By choosing one of our space-like co-ordinate directions parallel to S, we obtain T and L's co-ordinates as [h.c/E,0,0,0] and [0,0,0,h/q], with q = √g(p3,p3). Since K·T = 1 = K·S, we can now infer that h.K has, at least as to magnitude, time-like component E/c and space-like component equal to the space-like momentum, p3. Exactly the same is true of the 4-momentum, p: however, p is a vector where h.K should be a covector, and we only actually know the size of h.K's time-like and spatial components, not the directions. The geometry's metric, g, provides an isomorphism between vectors and covectors, so g(p) is suitably proportional to p but of the right type (i.e. a covector) to be a candidate for h.K's value; in our co-ordinates, g negates our time co-ordinate while preserving the spatial ones (just like a reflection in the surface time=0), so we obtain K = [−E/h/c,0,0,q/h] = g(p)/h.

Now, K annihilates [0,1,0,0], [0,0,1,0] and S−T = h.[–c/E,0,0,1/q], so a vector perpendicular to its kernel must be parallel to [E/c,0,0,q], i.e. to p. We know that K(p) = g(p,p)/h = m.m.c.c/h with m the rest-mass of our matter (i.e. its total energy, divided by c.c, in its proper frame of reference, whose time co-ordinate is the matter's proper time); whence K(h.p/g(p,p)) is one, making h.p/g(p,p) = h.p/(m.c)2 the natural displacement to describe as the period of the structure of our matter. In the matter's proper co-ordinates, this vector is simply [h/m/c,0,0,0], i.e. matter is periodic in the direction of its own proper time and a mass m has period h/m/c; this is a length (albeit in a time-like direction; feel free to divide by c again if you want it in units of time); the wave covector of matter with momentum p is K = g(p)/h.

This suffices as a start-point from which to motivate Schrödinger's equation.

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