The Copehagen Calculation

I set myself the challenge yesterday of showing that the semi-classical antenna picture of the hydrogen atom gave us the same physics as the Standard Model, the Copenhagen picture of quantum mechanics. Actually, it was more of a reverse-challenge: the idea was to find a single example whereby the antenna picture fails to give the correct physics. If classical electromagnetics is the miserable failure at the atomic level that everyone seems to say it is, it should be easy to find examples.

The spontaneous transition of hydrogen from the excited p-state to the ground s-state is as good a place as any to start. We can readily compare the predictions of the Standard Model with those of Maxwell's Equations. I haven't done the calculation yet, but we'll see how it goes. Let's open the show with the Copenhagen Version.

In Copenhagen, the hydrogen atom starts off in an excited state, and there is a certain probability that it will instantly jump to the ground state, emitting a photon in the process. Of course, this is wildly out of whack with what the semi-classical picture says; but that doesn't matter. What matters is the measurable outcome of an experiment; and in practise, the simplest way to measure this is to prepare a sample of hydrogen gas with a known energy and measure the radiation which it gives off at that transition frequency.

It has to be a pretty low energy, because we want to be dealing with essentially a two-state system. If we give it too much energy, then other higher-energy states start to become "populated", and the system gets more complicated. Let's say we have one million hydrogen atoms, and we give the system enough energy so that one percent of them are in the excited state. Then we can calculate how much they should radiate.

It's a pretty sophisticated calculation, but fortunately we have some help from Richard Fitzpatrick's excellent  farside website at the University of Texas. He starts off by computing what they call the dipole coupling between the ground state and the three p-states, taking the Bohr radius a0 as the reference:

The coupling looks a little different for the first two states, but that's just because of the way physicists define the states according to z-axis spin, as opposed to the so-called "vector states" of the chemist. A chemist's Px and Py state would be a sums and differences of the first two "physicists states", and with the normalization you get back the factor of root two that appears in the third expression.

It's encouraging to note that this business of calculating the dipole moment is very similar to what we're going to do later in the semi-classical analysis. Of course, we use different words to describe what we're doing. The Copenhagen people talk about something called "Fermi's Golden Rule" for calculating these transition rates. It's hard for me to understand how people are so quick to accept something because they call it a Golden Rule, or any kind or rule for that matter. What's going to be different about the semi-classical calculation is that I'm not going to tell you that such-and-such is the "rule". We're going to be able to relate to real physics at every step of the way. But I digress...

We note that the frequency of this transition is close to 2.5 x 10^15 Hz, giving a wavelength of 120 nanometers. Furthermore, the energy of the excited state is 3/4 Rydberg, 10.2 electron volts. Fitzpatrick then uses an equation derived earlier for the transition rate:

The "lifetime" of the p-state is evidently a nanosecond or two. What does this mean in terms of the physics we can measure? Let's recall that we had a sample of 1,000,000 hydrogen atoms, with one percent of them in the excited state. I'm putting the numbers together and I get a power output of very close to 10 microwatts. That seems awfully high for such a tiny sample, but remember the excited state has a very short lifetime indeed. And also, it's worth noting that the numbers we picked correspond to a sample at quite a high temperature. The Boltzmann factor of kT is only 0.026 eV at room temperature, and to get 1% of the atoms in the excited state (energy = 10.2 eV) we need the ratio E/kT to be the natural log of 100. This pushes the temperature up to around 5000 degrees if I'm doing it right. (EDIT: Now that I try again, it comes out to 24 000 degrees K. I don't know where I got 5000. In any case it's really hot.) So 10 microwatts it is, I suppose.

Tomorrow we'll see how the calculation looks if we treat the atoms as tiny classical antennas. Does anyone want to bet we get the same answer?

Who says energy comes in lumps?

Everyone says energy comes in lumps. That's the basic starting point of our whole paradigm. You'd have to be crazy to argue with it.

Call me crazy, but where are the lumps? The lumps are supposed to be necessary to derive the black-body spectrum, but just where do they come from? When we left off the other day, I was contrasting the Copenhagen interpretation with the Schroedinger interpretation. In Copenhagen, the atoms in a hot gas are each in some particular state of definite energy. According to certain probabilities, they may randomly absorb or emit a quantum of energy, thereby jumping from one state to another. Energy always appears in discrete lumps, or quanta.

In the Schroedinger picture, atom is surrounded by a charge of cloud which can have any shape at all. From our human perspective it is convenient for us to analyze those shapes as the sum of what are called eigenstates: those correspond to the definite energy states of Copenhagen. The equations show that any combination of eigenstates will generate an oscillating charge distribution, which will tend to radiate or absorb energy, depending on the state of the ambient electromagnetic field. The atoms are doing this all the time except in the exceptional circumstance when they happen to find themselves in a pure eigenstate. Only then does its charge distribution become stationary, and accordingly it stops exchanging energy with the surrounding field, if only for a moment.

How can these two radically different pictures give the same physics? The obvious answer is that they don't. Everyone knows that energy is absorbed or given off by atoms only in discrete quanta. The so-called Schroedinger picture, as I've described it, violates this fundamental rule. Therefore it must be wrong.

But why is it wrong? If it's so wrong, it should be easy to show where it falls down. Let's take a typical calculation. But what exactly should we calculate? It's easier said than done.

In situations like this, the thing to do is to back right down to the simplest possible case. In the Copenhagen Theory, you have something called the Einstein A and B coefficients. They give you the probability of an atom jumping from one state to another state. The Einstein A coefficitent has a particularly simple interpretation: it represesents an atom in an excited state which emits a photon and jumps to the ground state. For the s-p transition of hydrogen, we can actually calculate it. (Or we can just look it up.)

What does the Schroedinger picture tell us about this transition? Well, on the face of it, things don't look so good for Schroedinger. An atom in the pure p-state has a stationary electron cloud, so it doesn't radiate at all. It stays that way forever. This clearly contradicts what we know to be true from Copenhagen.

Or does it? Just how do we put an atom into a pure p-state, and how long do we have to watch it before it decays into the s-state? It's not obvious how we would do an experiment to verify the Copenhagen results. And yet we know it to be true. How do we know it? Because we take bulk measurements of the radiation intensity of a gas containing atoms in both states. Knowing the total energy of the gas, we know what percentage are in the excited state. Copenhagen predicts how often they decay, and knowing the amount of energy in each decay, we can calculate the average radiation. The experiment verifies that Copenhagen is correct.

But notice that this experiment doesn't tell us anything about individual atoms suddenly jumping from one state to another. That is a human spin that Copenhagen attaches to the physics. To compare the Schroedinger result, we ought to compare only the bottom-line physics. That means we have a gas with a certain amount of energy distributed between the ground state and the excited state, and there is a certain amount of electromagnetic energy associated with that gas. That's what we can measure, and that's what we have to calculate using the Schroedinger picture.

Let's say that 10% of the atoms are in the excited state. Of course, that's Copenhagen language: Schroedinger would have us say that each of the atoms is in a partially excited state, to the extent of 10%. Because of the mix between the s and p states, each atom has an oscillating charge distribution. This means it is a tiny antenna. The laws for calculating radiation from antennas are well known. We just need to figure out how much the atoms are oscillating, and apply the antenna equations.

Do you think we will get the same result as the Copenhagen calculation?

What's so special about pure states?

When I left off yesterday, I was talking about pure states and mixed states. There is a widespread point of view that in quantum mechanics, you can only observe a system in a pure state, never in a mixed state. People go a little farther than this: they say that in a gas of hydrogen atoms, there all the atoms are in a pure state: some of them in the ground state, and some of them in the higher excited states. Radiation occurs by means of something called a  quantum leap, whereby an atom instantly jumps from one state to another, and in the process either absorbs or emits a photon.

This picture dates back to the Bohr atom of 1915 with its planetary model. The Bohr atom became obsolete in 1926, with the advent of the Schroedinger equation. Schroedinger's atom had clouds of charge density instead of little orbiting charged particles. The planetary orbits of the Bohr atom were replaced by the so-called "pure states" of the Schroedinger atom.

What made the "pure states" special? Absoblutely nothing. Not in the Schroedinger atom at least. They were arguably special back in the Bohr atom because they were the only states. More importantly, they were special because they received a special exemption from Maxwell's Equations, the laws of electromagnetics. Contrary to those well-established laws, an electron in the Bohr atom was allowed to orbit forever without radiating, as long as it stayed in one of its allowed states. Bohr gave no explanation of how it could do this: that's just the way it was.

The Schroedinger atom was very different. If an atom didn't radiate it was simply because there was no oscillating charge to drive the radiation. No special exemption from Maxwell's Equations was needed. If the charges began to oscillate, Schroedinger's atom would do just what it was supposed to do according to Maxwell's Equations.

Then what were the so-called "pure states" all about? It's like this. Schroedinger wrote a differential equation to describe the hydrogen atom. There is a way to solve differential equations called "Separation of Variables". It's a funny method because you declare at the start that you are only going to look for solutions which can be broken down into the product of two functions, where one of them is a function only of space, and the other a function only of time. When you solve the equation this way, you get the so-called "pure states", also called eigenstates. There is really nothing special about them in a physical sense.

Are they the only states in which the atom can exist? Of course not. They came about only because you chose to restrict your mathematical solution to only those types of states. There is absolutely nothing in the physics which says that the wave function of hydrogen must be the mathematical product of a space function and a time function. That came from using the technique of Separation of Variables.

Why do people use this technique if it only gives you a tiny fraction of all possible solutions? There is method in the madness after all. Under the right circumstances, it turns out that all real physical solutions of the equation can be expressed as the mathematical sum of these special eigenstates. It may be a sum of just two or three of them, or it may be an infinite sum. It is especially useful to do this for things like the hydrogen atom because the time-dependence of the eigenstates is especially simple: they are all states of pure frequency. These kinds of summations are easy to handle mathematically, and they tend to have clear physical interpretations.

The physical picture for the hydrogen atom is especially satisfying. The Schroedinger equation ultimately tells us where the charge is distributed in the atom. It turns out that for the eigenstates, the charge distribution is stationary. That's why they are stable: because stationary charges don't radiate. Once you mix two states together, however, you find that there is an oscillating charge distribution. The frequencies of oscillation are exactly those that you measure in a heated gas by means of spectoscopy. It's obvious that the oscillating charges are responsible for the thermal radiation of a hot gas. 

So the question becomes: why do the adherents of the Copenhagen interpretation insist that they are not interested in the mixed states but only in the pure states? It's hard to say. Perhaps in the eleven short years when the Bohr atom held sway, those scientists got so used to thinking about special states and quantum leaps that they just couldn't bear to give it up, even when that point of view became totally unnecessary.
So they cobbled together their own interpretation of Schroedinger's equation, whereby his eigenstates were accorded special status. Unlike any other differential equation, they weren't going to allow you to construct new states by adding together eigenstates. Every atom was going to be in a pure eignestate, and instead of allowing the atoms to make a smooth transition from one eigenstate to another, they would calculate the probability of the atom jumping between states, just as they used to do with the Bohr atom.

So which picture is right, the Copenhagen interpretation or the Schroedinger picture. Incredibly, the Copenhagen people are able to show that their interpretation gives all the correct answers when applied to experiments. Doesn't this mean that they are right and Schroeginer is wrong?

No it doesn't. To do that they would have to show that you get the wrong answer if you treat the atoms as oscillating charge distributions and apply Maxwell's equations to calculate the resulting radiation. That's easier said than done. I know, I know, people are jumping up and down and shouting: "Ultraviolet catastrophe! Ultraviolet catastrophe!" There is no ultraviolet catastrophe with the Schroedinger atom. There is no ultraviolet catastrophe because the vibrations which would cause it are suppressed at the mechanical level.

Think about it: why wouldn't you get the correct black-body spectrum just by analyzing the classical radiation you would calculate from the charge oscillations as given by the Schroedinger equation? This is a straighforward question with a well-defined answer: either it's right or wrong. What is astonishing is that you could probably ask this question to 100 physicists, and not one of them would have ever thought of asking it. Oh, most of them would "know" the answer in the sense that they would "know" that you obviously get the wrong spectrum. But how do they know? They've never done the calculation, and it's not in any book. Everyone knows you're not allowed to use Maxwell's equations to calculate the radiation from quantum systems. That leads to nonsense. Quantum systems operate according to completely different laws.

Or do they?

Evolution or Intelligent Design?

The other day I said I had a problem with the theory of evolution. I broke the theory into two parts, which I called the Weak Theory and the Strong Theory. I said I was OK with the Weak Theory. That's the part that says we can trace our ancestry all the way back to the primordial slime. Who but an ignorant hillbilly could doubt that, in the light of the massive scientific evidence?

No, I'm not going to argue with the Weak Theory. It's the Strong Theory that gets me. I said I was going to hold off for a day or two and see if anyone could guess what I mean when I talk about the Strong Theory. Well, time's up.

The Weak Theory says,"this is exactly what happened". The Strong Theory says we know how and why it happened. The intricate development of millions of different life forms occurred through the mechanism of random errors in the transmission of the genetic code. The overwhelming majority of such transmission errors would naturally be harmful if not fatal; but here and there, it results in a characteristic which is useful. The bearer of this characteristic is more likely to procreate than his "normal" siblings, and the new characteristic is thereby incorporated into the gene pool of the population, where over time it ultimately dominates and becomes the new norm.

Is there one single scrap of scientific evidence to support the Strong Theory? In school we learned about the grey moths and the white moths. It seems that 99% of the moths used to be white, but during the Industrial Revolution smog coated everything in London with a layer of grey; the white moths became easy prey for birds, and as a result now all the moths are grey. Evolution at work.

But there were already grey moths in existence. The characteristic of interest was already present in the gene pool. Where was the random mutation that supposedly created it? There was none, because the gene already existed.

How does a brand new trait come into existence? That's a tough one. The much-maligned proponents of Intelligent Design like to talk about the problematic case of the human eye, and rightly so. There was once a time when not a single organism was sensitive to the presence or absence of light. They were all just blobs floating around in the ocean, and when one of them bumped into another, presumably the bigger one would wrap itself around the smaller one and mulch it down into digestible bits. A case of the blind eating the blind.

Now, what does the Strong Theory tell us? That somehow, one of these blobs laid its blob-eggs somewhere, and a cosmic particle came along and messed up the DNA in one of the eggs. The result was that a baby blob was born who was different from all the other blobs: somewhere on the surface of its body, there was a little patch of skin that contained photo-sensitive chemicals: compounds that would change state when exposed to light, and change back again when the light disappeared. Quite a trick for a random cosmic particle.

But that's not nearly enough. Somehow this patch of skin had to be hooked up to whatever rudimentary brain this blob must have possessed: the blob had to know that it was sensing light. The blog had to know that this tingling on its skin meant light, and it had to want to swim towards the source of that tingling, because perhaps there was more food to be had in that direction. By eating more food, the blob got bigger, and that made him more likely to survive in his blob-eat-blob world to the age of procreation. And his photo-sensitive offspring, over time, came to rule the world of the blobs through their inherent superiority.

So let's recap all the things that had to happen in a single generation for this genetic mutation to have taken hold. First, you needed the appearance of photo-sensitive compounds. This doesn't just happen by shooting a cosmic particle at a piece of DNA: the DNA doesn't become light sensitive. The DNA is a kind of a blueprint that creates an enzyme, and it is this new enzyme that manufactures the photo-sensitive chemical. But that's not enough. This new chemical, which is presumably just floating around in the protoplasm, has to somehow stimulate whatever locomotive mechanism this blob uses to propel itself around. It doesn't do any good to have a mutation which does one without the other.

That doesn't begin to address the question of how the blob is supposed to know whether it wants to swim towards the light or away from it. But never mind. Let's suppose it all works: that a single cosmic particle disrupts the genetic code in just the right way so that these three simultaneous, highly improbable developments occur (and without accidentally killing the blob in the bargain). The hard part is behind us; we've initiated the evolutionary chain, and now it's only a matter of time before this primitive photo-sensitive blob, the ancestor of us all, evolves into the myriad of life forms with the sophisticated organs of vision which we are all familiar with. Right?

The Creationists like to talk about a hurricane blowing through a junk yard, and when the dust has settled you find that by randomly flinging one piece of metal against another, it has managed  put together a Boeing 747. I'd say this is a pretty close to how the Strong Theory explains the evolution of the human eye. I know that saying this brands me as an ignorant hillbilly, but if the shoe fits I might as well wear it. I'd just like to throw one more wrinkle into the story of vision. It's the question of color.

Not all mammals have color vision. Presumably we humans have some ancestor who saw only black and white. What does the strong theory tell us? That somewhere along the line, a random mutation produced an individual with color vision, and because of his superior skills, his progeny came to dominate the gene pool over time.

A random mutation produced an individual with color vision? That's some cosmic particle we're talking about. Because anything less than a fully developed sense of color vision couldn't have been all that useful. Do you have any idea how much more complex a color TV is as compared to black and white? There are three different receptors, and there's all kinds of signal processing you've got to handle all of a sudden, and its a huge complication. And you can't argue that it came about gradually, because what kind of benefit is there to the individual who can see just a little red?

Come to think of it, how much of an advantage is color vision even when fully developed? Yes, it's nice to have, but it's not clear to me how an individual born out of the blue with the miraculous gift of color vision would have had all that much advantage over his black-and-white fellows. There were no color-coded electrical wires to worry about back then; yes, you could more easily tell the difference between a blackberry and a raspberry, but aren't they both delicious after all? I'm not saying it wouldn't have come in handy now and then, but to the extent that one such individual would ultimately dominate the whole gene pool? I just don't buy it.

The thing that bothers me the most about the whole evolution thing is that it's taken as a litmus test for admission into decent society. Question any aspect of the theory and you're branded as an ignorant hillbilly. Why is this?

I think it's because the proponents of evolution really want to use it as a weapon against religious faith. That's why the Strong Theory is so important to them. The Weak Theory is, after all, fully compatible with the idea that God has guided the process along with the intent of creating us in his image. But that's not good enough for the evolutionists. They demand that we acknowledge the random nature of the process, so that there is clearly no role for a Creator in their world-view. And if that means they have to argue for a Strong Theory that by its very nature cannot possibly have any scientific evidence to back it up, then so be it. 

Why people don't understand quantum mechanics

People don't understand quantum mechanics because they think that energy can only come in descrete lumps. This was an early interpretation of the old quantum mechanics, that has become solidly entrenched in the collective psyche even at the very highest levels. It is wrong and misleading in almost every instance where it is applied. The most imporant example of its misuse is in the case of emission and absorption of radiation between atoms.

Everything that atoms do in the context of thermal radiation can be understood by analyzing them as tiny antennas that work exactly the same as classical radio antennas. This is not some kind of crackpot idea: I know exactly what crackpot ideas are because I've taken a crackpot test and scored 155 on it. Seriously, I know exactly what I'm saying and I can back it up. If you understand this about atoms and radiation then all of quantum mechanics starts to make a little more sense.

The first thing you need to do is throw away the old planetary model of the atom. That was invented in 1915 by Bohr, and it became obsolete in 1926 when Schroedinger came along. Unfortunately, the image of the Bohr atom is burned into everyone's brain, thanks in no small measure to the efforts of the school system. The electron cloud of Schroedinger's atom is a vague and fuzzy concept for most people. Fortunately, there are some good animations of it available. I had a really nice one from the University of Saskatchewan website that I linked to last year  in my article on the crystal radio, but as I check it out right now it's giving me a hard time over my browser. So I'm not sure if it works. Here's another guy selling applets that make the same point; even if you don't buy it, you can see some nice animations on the web page. The point is that in quantum mechanics there is such a thing as a charge density, and if you take an atom in any kind of superposition of pure states, then you get an oscillating charge density. That's what you see in the applets, and there's nothing controversial about it.

The stumbling block for many people is the idea that you have an atom in a superposition of pure states. I can't count the number of times people have told me this is nonsense...people who should know better. They say a hydrogen atom, for example, is either in the ground state or the first excited state...never half and half. If it is in the presence of thermal radiation, it may absorb a photon and instantly jump from the lower state to the higher state, or it may emit a photon and jump the other way. That's the old "quantum leap" made popular by Neils Bohr, and made obsolete by the Schroedinger Equation. Or it should have been made obsolete, except that the proponents of the Copenhagen School adapted it to the new Schroedinger atom. Schroedinger developed his theory to show that there was a sensible way for the atom to make a smooth transition from one state to the other, and he was appalled when his theory was hijacked to support the quantum leap.

It's true that the Copenhagen school cobbled together a workable interpretation whereby only pure states exist, but the point is it's not a necessary interpretation. What people don't realize is that there is no experimental way to distinguish between the Copenhagen model of discreet states and quantum leaps, versus the Schroedinger model of mixed states and continuous transitions. This point was made really well by people who know all about these things in a stackexchange.com discussion which I link to in this article. The fact that you can analyze everything in terms of pure states and quantum leaps doesn't mean that you have to analyze it that way.

So what does the universe look like in Schroedinger's picture? Every atom is in some mixture of pure eigenstates, and if you look at the applets I mentioned earlier, it's easy to see that the charges are oscillating. As oscillating charges, they ought to radiate according to the classical laws of antenna theory; and that's where I hit the second wall. "Even if you're right about the oscillating charges..." (notice how they start with "even if", not even admitting what is clearly a mathematical consequence of the theory), "even if you're right about that, you're not allowed to use Maxwell's Equations to calculate the resulting radiation."

Not allowed? Why the hell not? There are two possibilities. Either you get the wrong answer for the black body radiation field, or you get the right answer. If you can show me that you get the wrong answer, then the game is over and I have to pack up my marbles and go home. But what if you get the right answer? Then what becomes of your quantization of energy into discrete lumps? Either way, it has nothing to do with what I'm supposedly allowed to do.

Let's leave it there for now. I'll come back to this before long. But I think I've also left some unfinished business about the theory of evolution...

What's Wrong with the Theory of Evolution?

There are two parts of the theory of evolution, which I'll call the Weak Theory and the Strong Theory. There's probably a pretty good case for the Weak Theory. It's the Strong Theory I have a problem with, and today I'm going to tell you why.

The Weak Theory tells us that we're all descended from lower animals. It's not too hard for us to believe we share an ancestor with monkeys. I don't know how good the fossil record really is on this. They used to talk about the quest for the "missing link"...the fossil specimen that would bridge the gap between them and us. I don't know how well that continuum has been filled in over the last hundred or so years, but let's give the theory the benefit of the doubt.

If you buy into the idea that we are related from the monkeys, then you have to allow that it can go back even farther. Were monkeys once tiny like groundhogs, or chipmunks? I suppose maybe they were. And aren't the chipmunks just overgrown rodents? Let's agree that there is a chain of descent.

If we accept that our ancestors were mice, then all the mammals have an equally good claim on that line. Which makes us cousins to lions and tigers, cows and horses, whatever. That's the gist of the Weak Theory. It actually explains a lot of things: we all have similar skeletal structures, we all have two eyes and two ears, fingers and toes, teeth, hearts that pump blood, etc.

But of course it doesn't stop there. The mice from which we are descended were in turn descended from lower forms. Once upon a time all animals were reptiles; the mouse and the frog and the snake are therefore distant cousins; the birds are another branch of the same family; and so on. The frog was once a fish; the fish was once a jellyfish; and the jellyfish was once a festering slime. And so we are told that there is a continuous line of descent, whereby we can trace our own ancestry to that same primordial slime.

This is the Theory of Evolution, and who but an ignorant hillbilly could possibly doubt it? Haven't we all been shown the evidence in the form of the fossil record, which clearly shows the continuous transformation of all these various forms of life from one to another? What, you haven't seen the fossils yourself? No matter; the scientists assure us that the case is well established, and that's good enough for me. Because all that I have described so far is the Weak Theory, and I don't have an issue with that. It's the Strong Theory that bothers me.

What do I mean by the Strong Theory? I'd like you to think about that, because it's a distinction that people seem to gloss over. I've never seen it explicitly discussed in quite the way I'm going to tackle it, and I even wonder if it's obvious where I'm going with this. So I think I'll let you mull it over for a day or so, and then continue where I left off.

Marty does some real thermodynamics

I've been promising to post this for about two years now and I guess now is as good a time as any. You go on the internet and get into physics arguments with people and sometimes you lose. But sometimes you win pretty big. This was undoubtedly my finest hour, and now I'm going to tell you about it.

It all goes back to my long-standing gripe with the idea of photons. One of the major arguments as to why light needs to be in the form of particles has to do with the photographic process. If light is a wave, then how come you can get a developable photographic image from a very distant star, when any reasonable calculation shows that there is no way to concentrate enough energy on a single atom of silver to reduce it from silver bromide to the metallic form.

I had been arguing that you didn't need photons because the appearance of silver flecks was thermodynamically spontaneous. It's an important point because people argue that you need the energy of a whole photon to drive a detection event, whereas I was arguing that you didn't. It's actually a little more complicated than that. There are two thresholds people worry about. One is the amount of energy to drive the overall process, and the other is the amount of energy to push an electron into the intermediate state. The latter is theoretically recoverable, but the former definitely needs to be accounted for. Even if the reduction of metallic silver is thermodynamically favored, the proponents of the photon theory still claim you need photons just for the activation energy, because no one has ever proposed a mechanism whereby you could recover this energy: that is, until I came up with the idea of Quantum Siphoning.

The problem is I turned out to be completely wrong. I had been arguing this point for about a week with a guy named SpectraCat when he finally came up with the numbers. The enthalpy of the reaction was 99 kilojoules per mole, and it was the wrong way. It was a crushing defeat for my argument. It wasn't just the activation energy: the complete process required a definite input of energy.

Incredibly, I was able to recover from this huge setback by means of an ingenious thermodynamic argument, which I am presenting below. It's odd that I happened to know enough thermodynamics to be able to make this argument. It never would have happened if not for a project I had been involved in as a junior engineer fresh out of university, thirty years previously. I talk about that in my series Karma and carbon monoxide.

Here is how I answered Spectracat's point about the enthalpy of the reaction.

You have presented the following equation in support of your position:

AgBr ----> Ag + ½ Br2 ΔH = +99 kJ/mol

You point out that the reaction is endothermic and therfore cannot procede without the input of energy from a photon.

It’s a good argument, and it should be a good argument because it’s my argument. I was the one who suggested, in the face of some initial ridicule, that we could settle the question by looking at the thermodynamics of the process. It turns out that I was right to look at the thermodynamics but wrong to think that we would settle the question in this way.

The Gibbs Free Energy is, of course, the parameter which normally tells us if a reaction procedes to the right or to the left. You have used the enthalpy instead in presenting your numbers; but no matter. The correction for the entropy is in any event rather small; only a little more than 3 kJ/mol in this case. Nothing decisive.

The important factor you have neglected is concentration. The Gibbs Free Energy equation gives us the change in free energy only when the reactants and their products are present in stoichiometric ratios. In the photographic process, a typical crystal may have, after being exposed to light, only a literal handful of silver atoms out of trillions. It is apparent that the silver and silver halide species are very far from their stoichiometric proportions, and therefore a more careful analysis is required.

The themodynamically correct method must be to treat the crystal as a solid solution of silver and silver bromide. In its initial state, the crystal is 100% AgBr. Is the formation of a single silver atom thermodynamically spontaneous, or does it require a net input of free energy?

For convenience, we will double the reaction to clear fractions:

2AgBr -------------> 2Ag + Br2 ΔG = +192 kJ

We can calculate the familiar equilibrium constant from first-year chemistry with the formula

K = exp(-ΔG/RT)

With RT = 2.2 kJ (approx) we get

With this information we can write the chemical equilibrium equation:

This equation easily solves for the equilibrium concentration of Ag being approximately one part in ten trillion (10^-13). This is within an order of magnitude or so of typical concentrations in an actual exposed crystal. It is apparent that for an initially pure crystal of AgBr, the spontaneous creation of trace amounts of silver is thermodynamically favored. The stimulation of the incident light wave (I’m going to stop calling it a photon) merely speeds up the process, but is not strictly necessary energetically.

We can do the same calculation a different way. Use the Gibbs Free Energy to calculate the conversion of one part per trillion (10^-12) of silver. It comes to 192 nanojoules. This is the input of free energy required to drive the process. But this assumes that the two species are unmixed. In fact, we ought to treat the silver as being in solution. Then we can show that, within the accuracy of this calculation, the needed energy is available from the entropy of mixing.

The entropy of mixing is given by the formula

ΔS = nRm*ln(m)

where n is the number of moles and m is concentration of the mixed species. Multiplication by T gives you the free energy of mixing:

ΔG = T ΔS = nRTm*ln(m)

For two moles at a concentration of one part per trillion, the free energy comes to 121 nanojoules; and make no mistake, it is in the right direction to drive the reaction forward. It is true that with the numbers I have chosen we are just a little short of the 192 nanojoules we said we needed, but remember we haven’t yet accounted for the contribution from the mixing of the bromine. In any case, you only need to drop the concentration another factor of ten to tilt the reaction decisively to the right.

The case becomes even more convincing (convincing to me, you understand: I know you're still not convinced) when we recall that the crystal in its pure form is considered to be a poor photodetector. In practise the material must be doctored by the addition of impurities, dislocations, and what are called “electron traps” to become really effective. It’s not hard to imagine (OK, for me to imagine) that the energy needed at the trace concentrations we are dealing with comes at least partly if not in large measure from the “doping” of the crystal. In other words, there is plenty of chemical energy available to drive the transition from silver bromide to silver without needing the energy of a photon. This is in line with my original description of the process as proceding from a metastable state to one of lower energy.

And that's why you don't need a photon to supply the energy to create a detection event. The energy needed is already available in the "detector" (the photographic plate)!

This still doesn't explain where you get the activation energy. The way we understand the process, a bound electron first has to get into the conduction band, from whence it is captured by a silver atom. At the time I wrote this analysis, I still didn't have an explanation of how that could happen. That's where Quantum Siphoning comes in, which brings my argument full circle. But I've already said that, haven't I?