Wednesday, March 17, 2010
“The collapse of the wave function” has been with us for eighty-some years (almost a hundred if we include its immediate precursor, the “quantum leap”). It has to be considered one of the great philosophical bugaboos (?) of quantum mechanics and modern physics in general. But what exactly is it? If we could possibly agree on the definition of wave function collapse, I imagine it would have to be something close to the following: that in a physical process, governed by laws of nature in the form of differential equations, where the state of a system (the wave function) evolves continously through time, there comes a moment when something happens to the wave function at one place, and simultaneously, the wave function everywhere else simply ceases to exist. It’s a paradox.
Perhaps there are people who would find fault with the above characterisation of wave function collapse. No matter; there are surely countless examples which we could draw on to demonstrate the phenomenon. Unfortunately, I have not been able to find anywhere a published list of the 10 or 100 most compelling examples of wave function collapse. Quite the contrary; the Wikipedia article for example contains not a single concrete instance (although it does link to an article on Schroedingers Cat). So I have no choice but to go out on a limb and choose my own example.
I choose to focus on the specks of silver which appear on a photographic plate when it is exposed to the weak light of a distant star. Few would dispute that at least in the mass culture, this example would be near the top of almost anybody’s list. But in precisely what manner does it so convincingly epitomize the concept? We will find that question is not so easy to answer.
Superficially, the argument probably goes something like this. The reduction of metallic silver from silver bromide takes a certain amount of energy. This energy can only come the light of the distant star. We can consider light in the form of electromagnetic waves, and easily calculate its power density at the location of the silver atom. It is easy to show that this density, calculated over the cross section of the atom, is far too low to account for the chemical reduction of silver in any realistic time frame. Ipso facto, collapse of the wave function.
This argument is shaky on the grounds that it uses classical electromagnetic light as the driving force. Classical wave functions don’t collapse; only quantum mechanical ones do. In quantum mechanics, light is made of point particles called photons. There is no philosophical problem with a photon striking a silver atom and driving the conversion; it’s just a matter of probability. No need to collapse a wave function.
The resolution is of course to combine the two arguments. Particle or not, the photon is still governed by a quantum mechanical wave function which is essentially the same as the classical wave function. Until a speck of silver appears on the plate, the wave function of the photon is spread out everywhere. Then, at the moment the speck appears, the wave function of the photon vanishes everywhere. It has to vanish everywhere because all of its energy was required at that one point in space in order to drive the conversion, of which the visible evidence is the residual fleck of silver.
But there is another way in which one can challenge the idea of wave function collapse in the case of the photographic plate. Is it really true that the reduction reduction of silver from silver bromide requires the full measure of energy from a single photon? At first glance, the question seems almost absurd in its naivety. The enthalpy of the chemical reaction
AgBr ------------> Ag + ½ Br2
is 99 kJ/mol, or nearly one electron volt per atom of silver, and it is in the positive direction, which means it requires the input of energy.
However, it turns out that the detailed physical chemistry of the process is somewhat more complicated than the simple one-line reaction written above, and even today it is probably not fully understood to the last detail. It is said that the role of crystal defects and trace impurities is critical in the efficient functioning of photographic film.
Furthermore, in calculating the free energy of the reaction from the enthalpy above, we must not forget to include the effect of concentration. A silver bromide crystal containing trillions of atoms may be developable after exposure even if it contains a mere handful of reduced silver atoms; and there is a fascinating thermodynamic argument based the on the entropy of mixing which shows that at such concentrations, the point of equilibrium shifts so far to the right as to make it at least arguably plausible (plausibly arguable?) that there may be locations within the crystal where the reduction of silver is in fact thermodynamically favored.
All this must surely seem highly speculative. But it is upon this faint hope which the argument to follow hinges. I am suggesting the possibility that the appearance of the silver fleck represents the transition of the crystal from a less stable to a more stable state energetically. And therefore it did not require the full energy of the photon to procede.
Even this radical assumption does not solve the problem of “collapse of the wave function”! Let us see why not.
The problem has to do with the energy barrier associated with the transition. In accordance with our proposed suspension of disbelief, we are going to suppose that the exposed AgBr crystal with its reduced silver atom is in a lower energy state than the undisturbed crystal. If this were the whole story, we would not need the energy of a photon to drive the transition. But the problem is in getting from A to B. According to our best understanding, the incident photon liberates a valence band electron by promoting it to the conduction band. I would like to say definitively how much energy this takes, but I have not been able to find this data. It is certainly a known fact that photographic film can be safely exposed to the deep red light of a dark room, but I haven’t found the actual cutoff frequency. Let’s assume it is 2 eV for the sake of argument.
The problem is that once we assume the photon is absorbed in promoting the electron to the valence band, we have implicitly assumed the collapse of the photon’s wave function. And that’s just what I’m trying to avoid.
The goal is to follow the process through the time evolution of the Schroedinger wave function to see if we can bring it to completion without at any stage invoking the collapse of the wave function. To facilitate this analysis, I have simplified the system to what I hope will be most easily manageable while retaining the essential features of the actual process.
What I have analyzed is the case of two potential wells inside a box. One well (A) is at a slightly higher potential than the other (B), and it is at (A) that the electron is trapped. Both wells are located inside a larger box (C) so the problem is confined to a finite volume.
So far, so good. But in what follows, it is a little tedious to have to remember which is A, which is B, and which is C. So I’m going to rename them to make things a little easier to follow. Remember, these are just names, and they aren’t to be taken as meaning any more than that.
We will refer to the well at A as “the silver halide”. The well at B, at a slightly lower potential , will be called “the silver atom”. And the big box, C, will be called “the conduction band”. Remember, despite what we call them, they are just two potential wells inside a box, and the similarities to any real chemical processes are, if not purely coincidence, at least incomplete. Oh, one more thing: the potential required to promote an electron from the “silver halide” to the “conduction band” (you see how this is going to work?) will be called “the band gap”.
We will assume that for one reason or another, the electron cannot get from the silver halide to the silver atom by tunneling. In our model we can arrange for this by having the two wells sufficiently far apart. We will further stipulate that the quantum of energy required to get into the conduction band is 2 eV.
What I proposed was that we consider what happens if instead of absorbing all the energy from a discrete photon, silver halide absorbs from a continuous electromagnetic field, only a fraction of that quantum of energy before the field disappears. Then the electron is then left in a superposition of states; say, 90% silver halide and 10% conduction band.. Since the conduction band is also coupled to the silver atom, the wave function may evolve further so that the electron finds itself in a superposition of 90% silver halide, 10% conduction band and 1% silver atom. (Due to rounding, percentages may not add to exactly 100.) And now the question: where does the system go from here? In particular, is it possible for the system to end up with the electron entirely in the silver atom?
Using the Born rule, standard QM would seem to tell us that there is a 90% probability of finding it in the silver halide, a 10% chance of finding in the conduction band, and a 1% chance of finding it in the silver atom. But that is not especially helpful. For one thing, as long as the conduction band is partially filled, there will be radiating energy which we are going to assume can escape from the box. And as the radiant energy is lost from the system, the conduction band will be depleted to the benefit of both the silver halide and the silver. We might then expect the system to stabilize in the proportions of 90/10 silver halide to silver. The exact percentages don’t matter; the point is, there is a probability of finding the electron in the silver atom.
But the Born rule which we used to get those probabilities essentially demands the collapse of the wave function, and that’s just the thing we’re trying to avoid. We avoided it at the point the photon was absorbed by saying we wouldn’t take a whole photon, we’d just take a portion of a continuous e-m wave. It defeats our whole purpose to now invoke the Born Rule to force it into the silver atom. We return to the question: can the electron get from the mixed state of silver/silver halide/condutcion band to a state where it is entirely in the silver atom, and can it get there by means of a process which evolves naturally in space and time?!!
There is enough energy available to drive the process; that’s not the problem. When the conduction band couples to the silver atom, it is true that the probability drains toward the silver which is what we want. And energy is released which we would like to use in order to replenish the silver halide, to keep the process moving. Very much like an ordinary siphon. The probelm is that the energy at the silver atom is released in the form of radiating electromagnetic energy, and it was not clear to me how could I recapture it with good efficiency at the silver halide, especially if the two are relatively far apart.
I though about this for a long time and couldn’t get around it. Then the answer came to me. You can’t do it. You can’t recapture the energy lost radiated away from the silver atom, receive it at the silver halide, and pump it back up to the conduction band.
But you can do it if you have millions and millions of silver halide sites! We will call such a collection of potential wells (remember, that’s all they are) a “crystal”. In the very middle of the crystal is one special well, the “silver atom”, just a little deeper than all the other wells, and it is empty.
Now pass a wave of light through the “crystal”. Not a very strong wave, but at a frequency sufficient to couple the silver halides to the conduction band. When the wave is gone, the conduction band is excited to the extent of 10%, and all the silver halides ground state wave functions are filled to the extent of only 99.99999%.
Notice carefully that according to my picture, you haven’t captured enough energy to put a whole electron into the conduction band. Just 10% of one. And yet now I’m going to show how the system will evolve through time so that the electron ends up in the silver atom. The process is what I call quantum siphoning.
It’s not hard to get a little bit of wave function excitation happening in the silver atom. There’s plenty available in the conduction band to excite it to the level of, say 1%. But as it starts to fills up, the conduction band is emptying out, and it is not being replenished. The energy which was available is simply being radiated away from the site of the silver atom.
But the silver atom is surrounded by millions of receiving antennas which are tuned to its exact frequency. These are none other than all the surrounding silver halides. As the electromagnetic wave radiates outwards, it can’t help but excite the millions of potential wells in its path. And as it does, it leaves a little of its energy behind at each one, getting weaker and weaker as it goes. At the same time a wee liitle portion of all the electron wave functions in those millions of silver halides is is promoted to the conduction band. In fact, the penetration depth of the em wave emanating from the silver atom decays exponentially, until essentially one hundred percent of it is absorbed. And every bit of that absorbed energy goes towards replenishing the conduction band. There is literally nowhere else for it to go. It is in fact the quantum mechanical version of a siphon. The replenished conduction band can now continue to refill the silver atom ground state until it’s completely full. Full of exactly one electron.
Let us recap: A weak, diffuse electromagnetic wave passed through the crystal. A small portion of a quantum of energy was absorbed in the passage, and the effect of this absorption was recorded by the appearance of an electron in the silver atom. Every step of the process occured according to the ordinary time evolution laws of the Schroedinger and Maxwell equations, and at no point was it necessary for a wave to “collapse to an eigenfunction of the state being measured.”
And finally, not to argue the point to strongly, although it is just a though experiment with potential wells scattered in a big box, it does seem to shares some of the characteristic features of the traditional photographic plate, with its flecks of silver appearing one by one under the influence of very weak starlight. Is light made of photons? Does the wave function collapse? Or can we not perhaps explain it all by wave-on-wave interactions according to the well-known laws of Schroedinger and Maxwell?