Tag Archives: Dr. Who

Dr. Who’s Quantum reality viewed as diffusion

It’s very hard to get the meaning of life from science because reality is very strange, Further, science is mathematical, and the math relations for reality can be re-arranged. One arrangement of the terms will suggest a version of causality, while another will suggest a different causality. As Dr. Who points out, in non-linear, non-objective terms, there’s no causality, but rather a wibbly-wobbely ball of timey-wimey stuff.

Time as a ball of wibblely wobbly timey wimey stuff.

Reality is a ball of  timey wimpy stuff, Dr. Who.

To this end, I’ll provide my favorite way of looking at the timey-wimey way of the world by rearranging the equations of quantum mechanics into a sort of diffusion. It’s not the diffusion of something you’re quite familiar with, but rather a timey-wimey wave-stuff referred to as Ψ. It’s part real and part imaginary, and the only relationship between ψ and life is that the chance of finding something somewhere is proportional Ψ*|Ψ. The diffusion of this half-imaginary stuff is the underpinning of reality — if viewed in a certain way.

First let’s consider the steady diffusion of a normal (un-quantum) material. If there is a lot of it, like when there’s perfume off of a prima donna, you can say that N = -D dc/dx where N is the flux of perfume (molecules per minute per area), dc/dx is a concentration gradient (there’s more perfume near her than near you), and D is a diffusivity, a number related to the mobility of those perfume molecules. 

We can further generalize the diffusion of an ordinary material for a case where concentration varies with time because of reaction or a difference between the in-rate and the out rate, with reaction added as a secondary accumulator, we can write: dc/dt = reaction + dN/dx = reaction + D d2c/dx2. For a first order reaction, for example radioactive decay, reaction = -ßc, and 

dc/dt = -ßc + D d2c/dx2               (1)

where ß is the radioactive decay constant of the material whose concentration is c.

Viewed in a certain way, the most relevant equation for reality, the time-dependent Schrödinger wave equation (semi-derived here), fits into the same diffusion-reaction form:

dΨ/dt = – 2iπV/h Ψ + hi/4πm d2Ψ/dx               (2)

Instead of reality involving the motion of a real material (perfume, radioactive radon, etc.) with a real concentration, c, in this relation, the material can not be sensed directly, and the concentration, Ψ, is semi -imaginary. Here, h is plank’s constant, i is the imaginary number, √-1, m is the mass of the real material, and V is potential energy. When dealing with reactions or charged materials, it’s relevant that V will vary with position (e.g. electrons’ energy is lower when they are near protons). The diffusivity term here is imaginary, hi/4πm, but that’s OK, Ψ is part imaginary, and we’d expect that potential energy is something of a destroyer of Ψ: the likelihood of finding something at a spot goes down where the energy is high.

The form of this diffusion is linear, a mathematical term that refers to equations where solution that works for Ψ will also work for 2Ψ. Generally speaking linear solutions have exp() terms in them, and that’s especially likely here as the only place where you see a time term is on the left. For most cases we can say that

Ψ = ψ exp(-2iπE/h)t               (3)

where ψ is not a function of anything but x (space) and E is the energy of the thing whose behavior is described by Ψ. If you take the derivative of equation 3 this with respect to time, t, you get

dΨ/dt = ψ (-2iπE/h) exp(-2iπE/h)t = (-2iπE/h)Ψ.               (4)

If you insert this into equation 2, you’ll notice that the form of the first term is now identical to the second, with energy appearing identically in both terms. Divide now by exp(-2iπE/h)t, and you get the following equation:

(E-V) ψ =  -h2/8π2m d2ψ/dx2                      (5)

where ψ can be thought of as the physical concentration in space of the timey-wimey stuff. ψ is still wibbly-wobbley, but no longer timey-wimey. Now ψ- squared is the likelihood of finding the stuff somewhere at any time, and E, the energy of the thing. For most things in normal conditions, E is quantized and equals approximately kT. That is E of the thing equals, typically, a quantized energy state that’s nearly Boltzmann’s constant times temperature.

You now want to check that the approximation in equation 3-5 was legitimate. You do this by checking if the length-scale implicit in exp(-2iπE/h)t is small relative to the length-scales of the action. If it is (and it usually is), You are free to solve for ψ at any E and V using normal mathematics, by analytic or digital means, for example this way. ψ will be wibbely-wobbely but won’t be timey-wimey. That is, the space behavior of the thing will be peculiar with the item in forbidden locations, but there won’t be time reversal. For time reversal, you need small space features (like here) or entanglement.

Equation 5 can be considered a simple steady state diffusion equation. The stuff whose concentration is ψ is created wherever E is greater than V, and is destroyed wherever V is greater than E. The stuff then continuously diffuses from the former area to the latter establishing a time-independent concentration profile. E is quantized (can only be some specific values) since matter can never be created of destroyed, and it is only at specific values of E that this happens in Equation 5. For a particle in a flat box, E and ψ are found, typically, by realizing that the format of ψ must be a sin function (and ignoring an infinity). For more complex potential energy surfaces, it’s best to use a matrix solution for ψ along with non-continuous calculous. This avoids the infinity, and is a lot more flexible besides.

When you detect a material in some spot, you can imagine that the space- function ψ collapses, but even that isn’t clear as you can never know the position and velocity of a thing simultaneously, so doesn’t collapse all that much. And as for what the stuff is that diffuses and has concentration ψ, no-one knows, but it behaves like a stuff. And as to why it diffuses, perhaps it’s jiggled by unseen photons. I don’t know if this is what happens, but it’s a way I often choose to imagine reality — a moving, unseen material with real and imaginary (spiritual ?) parts, whose concentration, ψ, is related to experience, but not directly experienced.

This is not the only way the equations can be rearranged. Another way of thinking of things is as the sum of path integrals — an approach that appears to me as a many-world version, with fixed-points in time (another Dr Who feature). In this view, every object takes every path possible between these points, and reality as the sum of all the versions, including some that have time reversals. Richard Feynman explains this path integral approach here. If it doesn’t make more sense than my version, that’s OK. There is no version of the quantum equations that will make total, rational sense. All the true ones are mathematically equivalent — totally equal, but differ in the “meaning”. That is, if you were to impose meaning on the math terms, the meaning would be totally different. That’s not to say that all explanations are equally valid — most versions are totally wrong, but there are many, equally valid math version to fit many, equally valid religious or philosophic world views. The various religions, I think, are uncomfortable with having so many completely different views being totally equal because (as I understand it) each wants exclusive ownership of truth. Since this is never so for math, I claim religion is the opposite of science. Religion is trying to find The Meaning of life, and science is trying to match experiential truth — and ideally useful truth; knowing the meaning of life isn’t that useful in a knife fight.

Dr. Robert E. Buxbaum, July 9, 2014. If nothing else, you now perhaps understand Dr. Who more than you did previously. If you liked this, see here for a view of political happiness in terms of the thermodynamics of free-energy minimization.