Sunday, February 19, 2012

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?

No comments: