## 19 The wave equation

Schrödinger’s equation looks like this:

Schrödinger did not create this in the way that Maxwell did with his equations. This is a brilliant piece of pure mathematical modelling. The key element here is ψ (or psi). It is shown as varying in both space and time (x and t in brackets), and in quantum theory ψ does not have a physical interpretation, although the square of ψ is a measure of the probability of a particle being observed.

Schrödinger’s equation is known as a wave equation, though not to Schrödinger himself, as we have seen. The reason Schrödinger distanced himself from the appellation is not known, but it might well have been related to the fact that it has the wrong structure to be a wave equation. The classic wave equation looks like this:

The key difference, and a fundamental one, is in the derivatives, shown both as a curly ‘d’ and an inverted triangle. A derivative in mathematics is a gradient. More straightforwardly, it is a rate of change. Velocity, for example, is a derivative, as it is the rate of change of position as time changes. We might represent that by dx/dt. We see t representing time on the bottom of derivatives in both equations. Values of pressure or velocity might also change from place to place, and this is shown either by an x in the denominator, or the del symbol (an upside down ∆, Greek ‘delta’).

The upper case ^{2} on the d or curly-d or del indicates a ‘second derivative’ or rate of change of a rate of change, for example as acceleration is to velocity, and this is a crucial difference between the equations. Both equations have a second derivative with respect to position, but one, the Schrödinger equation, has a first derivative with respect to time, while the wave equation has a second derivative.

This is fundamental. The Schrödinger equation is not a wave equation. This is not a break with established ‘knowledge’ as we had with the transverse electromagnetic wave. Physicists know that this is not a wave equation; they just call it that. As obfuscation in physics goes, this is a minor transgression, and we will let it go. As someone important once said, ‘they know not what they do’.

Here is another equation in everyday use at the heart of an essential science:

This contains u as the variable, the key subject of the equation, but in this case it represents velocity, whereas in the classic wave equation above it represents position. Both equations therefore have a term representing acceleration. However, in the wave equation it is a second derivative of position (u) and in this equation, known as the Navier-Stokes equation, it is the first derivative of velocity (disturbingly also u). Both equations are core equations of motion. As with Newton’s most famous equation, F=ma, core equations of motion tend to have a term representing acceleration at their heart.

There is some other stuff in each of our three equations, but the key to understanding the Schrödinger equation is in the derivatives.

The derivative structure of the Schrödinger equation is a match, not to the classic wave equation, but to the Navier-Stokes equation, and the Navier-Stokes equation is a core equation of hydrodynamics. This allows us, with some confidence, to conclude the following:

The Schrödinger equation is a hydrodynamic equation.

It is a version of the Navier-Stokes equation of hydrodynamics.

Schrödinger’s ψ is velocity.

And we haven’t finished yet!

Schrödinger’s ψ is a complex term, not just in the common usage of that word but also in the mathematical one:

It involves complex numbers, wholly invented numbers involving the symbol i, being the non-existent (and hence ‘imaginary’) square root of minus one.

Once we know that the Schrödinger equation is hydrodynamic in nature, rather than an abstract piece of pure mathematics, and that ψ represents velocity, we can also interpret this strange use of imaginary numbers in a simple physical way. We do not need to delve into the complexities of complex numbers at all, merely to know that they are commonly used to represent rotation, and that is what they are doing in this equation.

The Schrödinger equation of quantum mechanics, the core equation of that discipline, is a hydrodynamic equation that very powerfully and effectively describes rotation in the aether, Maxwell’s aether, confirmed as fluidic. It tells us that the structures it is designed to model, the hydrogen atom and the component parts of larger atoms, are rotational structures in a fluidic medium.

And it does a little more. When we look at ψ as a velocity, we (or at least the mathematicians among us) can see that the velocity is orthogonal to (at ninety degrees to) both the acceleration and the radius (giving the position), and this is very specific to circular rotation where we know the velocity is tangential and there is a restoring force directed towards the centre. The reader with any kind of technological background will realise that what we are describing is a circular vortex.

There are further theoretical precedents for this. Hydrodynamic explanations for phenomena covered by twentieth century theory date back well before 1900. Max Jammer states that ‘The earliest hydrodynamic interpretation was proposed by Erwin Madelung’, and that ‘Arthur Korn … had published in 1892 a hydrodynamic theory of gravitation and of electricity which he subsequently extended to optics and spectroscopy.’ [i]

[i] Jammer, pages 33 & 37