## Sunday, October 2, 2011

### The Collapse of the Wave Function

I didn't know when I started this blog two years ago that within a few months I would figure out how to explain one of the most baffling phenomena of quantum mechanics: the collapse of the wave function. I've already written it up under the title of "Quantum Siphoning" but in that article I dealt in detail with the actual mechanism. Today I'm going to revisit the "big picture".

I've taken as the prototypical example of wave function collapse to be the appearance of flecks of silver on a photographic plate when exposed to the weak light of a distant star. The reduction of silver bromide is a chemical process that requires a significant input of energy. You do not get a silver fleck on the plate without the reduction of silver bromide. So where does the energy come from? Obviously, it must come from the light wave. But it is a fact that we can make the light source weaker and weaker without limit, and yet all we have to do in order to see flecks appear is to wait long enough. How can we explain this unless we accept that the energy of the light, contrary to Maxwell's Laws, is present in concentrated lumps?

The answer to this paradox is suttle and unexpected. We must consider the reduction of silver bromide from a thermodynamic perspective. In thermodynamics we do not talk about A and B reacting to form C; we say that A, B, and C are in equilibrium; and that equilibrium occurs when the rate of combination of A and B into C is equal to the rate of decomposition of C into A and B. Everything is in flux.

We are taught to calculate the point of equilibrium by means of the Gibbs Free Energy Function. How does this apply to a photographic film? Here we must consider the equilibrium between silver bromide and metallic silver. The Gibbs Free Energy function tells us that it takes a great amount of energy to convert silver bromide to metallic silver. But that is only part of the story. The Gibbs Free Energy is calculated for reactants and products in their stoichiometric proportions: in this case, equal molar fractions of silver bromide and metallic silver. In fact, this is far from the case in an unexposed photographic plate.

In an unexposed film there is virtually no metallic silver; it takes only a few parts per billion of silver in a single crystal to render a developable image. To calculate the thermodynamics, we must then consider the crystal as a solid solution of reactants and products; and when the Gibbs Free Energy is recalculated taking this into account, it turns out that the equilibrium point is indeed on the order of parts per billion.

What does this mean? In equilibrium, the forward and backward conversion rates are equal. Silver is turning into silver bromide, and silver bromide is turning back into silver of its own accord. There is no net input of energy required to drive either the forward or reverse process. When a silver bromide atom turns to silver, or vice versa, there is no net change in the energy of the entire crystal.

If this is the case, then there is no valid argument whereby the energy of light must be delivered in concentrated lumps in order to drive the reaction. The conversion may be readily catalyzed by the smallest amount of light energy, because the energy needed for the transition is already present in the crystal.

I have worked out the mathematics of this and posted my calculation in this discussion on physicsforums.com. I'm going to repost it on my blog later, but for now you can check it out as post 44 in the discussion. The funny thing was that I wasn't really prepared to make this point when it came up. I had originally been arguing that the reduction of silver was spontaneous in the "ordinary" sense of having a negative free energy by the simple (stoichiometric) calculation, and I was shocked to find that I was wrong about that.
The reason I was able to recover was that twenty-five years earlier, as a young engineer, I had been exposed to a peculiar situation where a hydrocarbon detector was giving odd results. It was not showing the expected presence of methane in CO2, and I put forward the theory that the methane and CO2 were reacting to form carbon monoxide. I was ridiculed for this because the Gibbs Free Energy of my reaction was obviously positive, so everyone assumed it couldn't move forward. What they ignored was the effect of concentration. I was able to show that at the trace concentrations we were looking at (parts per million) the point of equilibrium actually tipped in the opposite direction. (There was also an elevated temperature to consider.) I've described this in a series of posts from earlier this year.
The point is that its the exact same thermodynamic calculation that turns the traditional explanation of "wave function collapse" on its head. The notion of collapse is based on the assumption that the energy for the process must have come from the photon, and I am able to show that the energy is already present in the detection system (the photographic plate.) If it's true in this prototypical instance of wave function collapse, then how many other situations might also be analyzed in this way?