Quantum Principles

This is where we find real randomness, not like the coin toss where the outcome is really determined by the nature of the flip and catch, but built-in randomness of nature (we think).

It all starts with the photon, and we find that the photon has energy hf = hc/l and momentum = h/l , where h is Planck's constant, 6.63x10-34 J s, f is frequency, l is wavelength and c is velocity of light. Sometimes photon energies are given in electron-volts, eV. It is then useful to know that the product hc is
hc = 1.24 eV× m m = 1240 eV×nm.

The photon is a quantum of light: an indivisible chunk which gives light its particle nature; thus light has both wave properties and particle properties. The particle nature reveals itself in the photoelectric effect, in which photons bounce electrons out of a metal; the Compton effect, playing billiards with photons; pair production, in which the photon gives birth and disappears. (There is also pear production in orchards, due to photosynthesis; and high intensity beams of photons can pare things down by burning or vaporizing. Thus photons are involved in pair, pear, and pare production.)

Long after the photon was discovered, Louis de Broglie made a brilliant and bold proposal in his dissertation: that if nature plays these spooky games with light, maybe the electron is also both a wave and a particle. He proceeded to show that Bohr's planetary model of the hydrogen atom, which had been cooked up ten years earlier to fit the data, could now be explained very nicely. Before de Broglie, Bohr's theory did not make sense. In de Broglie's view, the possible electron orbits were those which were standing waves. That is, the circumference of orbit, 2p r, must be a multiple of the wavelength of the electron: 2p r=nl , with n = 1,2,3,…

By analogy with the photon (see the top of this page), the electron has momentum h/l, so l = h/mv. We use conventional physics the rest of the way (force of attraction between electron and nucleus by Coulomb's law, = mv2/r for circular motion…), and we find that the electron energies = a negative constant divided by n2. Converting to electron volts, we find that

En = -13.6eV/n2 is the total energy of the nth orbit.

You can find the above derivation in a modern physics book or look here.

This works with the hydrogen atom, correctly predicting the possible electron energies. An electron in an atom can get "excited" to a higher than normal energy (higher n) by heat, light, collision with a particle (for example electrons due to the passage of electric current through the gas). Then a photon is emitted by the excited electron as it drops down to a lower orbit, and the drop in energy of the electron is the photon energy, hc/l . For problems involving this it is handy to know that hc = 1.24 eV mm = 1240 eV nm. The nanometer, 10-9 m, is commonly used with light. Visible light has l = 400 to 700 nm, approximately.

(We now know that the electron wave structure in atoms is governed by a three dimensional wave equation, Schrodinger's equation, which built upon de Broglie's success, but that is a long story.)

The committee at the university didn't want to accept de Broglie's dissertation because they didn't believe his theory, but the chairman sent a copy to Einstein. He liked it, so they passed de Broglie. How embarrassing it would have been for them had they not passed him, because he later was awarded the Nobel Prize for his theory!

For a free electron with non-relativistic kinetic energy E in electron volts,
l = [1.226 (eV)1/2 nm]/E1/2, but on tests, use this only as a check. Your instructor wants you to do it from basic principles: Convert E to Joules, then p = mv = (2mE)1/2 and l=h/p, where h=6.63x10-34 J s. This will give you l in meters.

Wave-particle duality:
We need to accept the overwhelming evidence that waves have a particle nature and the elementary particles have a wave nature. I think that most physicists do not attempt to make sense out of this, they just knuckle under to the fact that sometimes you need to treat them as waves and sometimes as particles, and I think that most of them would say they are not both at the same time. I will take the contrary view, but I doubt that there is any experimental difference between the two ways of looking at it, so maybe it is just one of those pseudo-philosophical puzzles, like when a man speaks his mind in the forest and there is no woman there to hear him, is he still wrong? For a more expert advocacy of the view that the thing is both a wave and a particle at the same time, see Sheldon Goldstein's two-part article in the March and April 1998 issues of Physics Today.

First let's look at the double slit experiment, in which all of the mysteries of wave-particle duality come out.

Review the double slit first if you don't remember how it works. It does not matter if we use light or electrons, the pattern we obtain on the screen is determined by the wave theory: dsinq = ml determines where the maxima occur. Yet if you send one photon or one electron at a time toward the slits, common sense (a dangerous thing to rely on) says that it goes through one slit or the other. (We can detect a single photon or electron, and when detectors are placed right behind the slits we confirm that it went through one or the other. Also, no one has ever found half a photon or half an electron.) The amazing thing is that sending one at a time does not change the pattern: the individual hits on the screen seem random at first, but after a long time the usual double-slit pattern emerges. So if the particle went through only one slit, how did it "know" about the other slit? Either it was a wave only at the time of passage through the slits and becomes a particle when detected, or it is always both. The sketch below shows the double slit, and the wiggly line on the right is a graph of intensity vs location on screen. The dotted line at angle q below center is drawn to the 2nd maximum (m=2).

The seemingly impossible really happens:
(Back to the double slit later.) Nature defies common sense, and this really happens: an electron or other particle can disappear in one location and simultaneously reappear at another location if the wave associated with the particle extends to the other location. After you buy that, we can make sense out of the rest of it. Here are some examples of the disappearing-reappearing magic:

Making sense out of the double slit and other mysterious things:
Most physicists say that it went through both slits as a wave, and its particle nature is not there until it is detected. All you need to make sense out of it is the tunneling-like magic. You do not need the additional magic of the particle nature being absent as the wave passes through the double slit. You can say that a particle going toward a double slit may go through one or the other, but its wave goes through both. The particle can jump around anywhere in its wave, and the square of the wave is the probability of finding it in any location. Is it doing all this jumping around as it travels through space? It doesn't hurt to picture it that way, but all we can say from the evidence is that the probability of its location where it will be detected is governed by its wave. (And of course its wave nature immediately changes upon detection- the "collapse of the wave function." The wave is nature's expression of the randomness, and the randomness of its location necessarily disappears when it is detected.)

Now you might wonder, "Waves of what?" Nobody has come up with anything better than probability waves. Thus it seems like they don't exist at all as real waves- probability is just a mathematical function. Yet the theory works, correctly predicting the results of countless experiments, and if the waves don't exist it is difficult to see how nature could behave the way it does. On the other hand, we would be stupid to fall into the Aristotelian trap of deciding how nature works just by thought. If nature plays randomness games, and if the probabilities are found by playing with non-existent waves, so be it.

The Heisenberg uncertainty principle deals with the built-in randomness of nature caused by the ability of the particle to locate itself anywhere in its wave.

If you are starting in a quantum mechanics class (or planning to take the course), you might look at my QM of the Crap Shoot, in which I define a state vector ď y > and eigenvalues, eigenvectors for the crap game (rolling a pair of dice), and show a few manipulations of these things. This will make life easier for you in QM, I hope. Then look at this, in which I attempt to show why the matrix mechanics approach and the wave mechanics approach are equivalent.

Another approach is Feynman's sum over paths theory, which works because the wave takes all possible paths (or nature acts as if the wave takes all possible paths). So there are three forms of the quantum theory, wave mechanics, matrix mechanics and Feynman's, and you use the one that is easiest or most efficient for the particular problem.

1… 2… 3… 4… 5