Study Guide for Exam I
Study Guide for Exam II
Quanta Triumphant
God Rolls Dice
Of Cats and Computers
Einstein was thought to be a bit eccentric for his quantum (photon) idea, although it's what he eventually got the 1921 Nobel Prize for (and not relativity). Everything physicists had done for 100 years indicated that light is an electromagnetic wave, not a particle. As a prime example of this, Robert Millikan, an American, carried out a series of highly precise experiments on the photoelectric effect, in which he was able to verify Einstein's 1905 prediction that the increase in stopping voltage as the light's frequency increased was not just any kind of increase but was a straight-line increase on a graph. Yet when he gave his Nobel Prize lecture, Millikan couldn't bring himself to accept that Einstein's idea was correct: He stated that his experiments had verified Einstein's equation but that the quantum concept that had led Einstein to that equation just couldn't be right.
But things started looking up for the quantum hypothesis in 1923, when another American named Arthur Compton shined X-rays at various materials such as iron and graphite; electrons in these materials scattered the X-rays and increased the X-rays' wavelength in the process. This made perfect sense if light is particles: the X-ray photons reached the material, "banged into" electrons like a cue ball on a break, sent the electrons flying, and lost energy in the process. Since lower energy photons correspond to lower frequency light waves, and lower frequency light waves correspond to longer wavelength light waves, it works.
So strange was the quantum idea, however, that Niels Bohr and two younger collaborators came up with the Bohr-Kramers-Slater (BKS) theory as a way out: Bohr was willing to jettison the principle of conservation of energy in order to avoid supposing that light is a stream of particles. You should know a bit about the BKS theory, and you especially should know something about the two 1924 experiments that finally clinched it for Einstein and the quantum.
At about this time, Einstein received a manuscript (in English) from an unknown physics professor from India, Satyendranath Bose, in which Bose treated light as quanta and described the behavior of these quanta in terms of the statistical laws they obey. (Note that Einstein's 1905 work on Brownian motion was the same kind of statistical analysis, but for atoms rather than light quanta.) Several physics journals had rejected Bose's article, but Einstein was so impressed that he translated it into German, submitted it to a leading journal for publication, and eventually extended Bose's work to consider particles that have mass but that obey the same statistical laws as do massless quanta.
Today we would say that particles that obey these "Bose-Einstein statistics" are called "bosons" and are defined by the fact that their "spin" has, in the tiny units used by physicists, an integer value. So, for example, a photon has a spin of 1, while a helium-4 nucleus has a spin of 0. Electrons, protons, and neutrons, on the other hand, have 1/2 unit of spin, so they are not bosons but are instead called "fermions" (after the Italian physicist Enrico Fermi).
And how do bosons behave? For our purposes, the important point is that they "like" to be in exactly the same state as each other. Fermions, on the other hand, are forbidden to be in the same state as each other; that's why you can only fit but so many electrons in each electron "shell" surrounding an atom, because each electron must differ in some way from each other electron in the atom. This is the basis of chemistry.
Bose-Einstein statistics are behind the concept of stimulated emission, and stimulated emission is the basic concept behind the laser. Consider an atom in a certain allowed energy level, and suppose that there's another allowed energy level 2.3 eV below that one. It turns out that the atom is more likely to make this 2.3 eV downward jump if there's a 2.3 eV photon passing through the neighborhood. This 2.3 eV photon, in other words, stimulates the atom to emit a second photon exactly like itself: same energy, same direction, same "phase" (i.e., exactly in synch with the first). This makes sense if you remember that Nature "likes" to have identical photons, since photons are bosons. The next excited atom encountered is even more likely to emit light, since it is being stimulated by two identical photons and not just one. If you can arrange for lots of atoms to start out in the upper state rather than the lower state -- a "population inversion" -- you can get a burst of identical photons ("coherent light") as the atoms all drop downward in a sort of chain reaction: each newly emitted photon makes it that much more likely that the next atom will drop downward in energy and emit a photon. The result is an intense light beam, some of which escapes through a thinly silvered mirror at one end of the laser.
A lot of folks who read about the history of science wonder why a brilliant guy like Einstein spent the last three decades of his life fighting unsuccessfully against quantum physics. But if you think about the story so far, it's not such a mystery. His 1905 work on special relativity and the existence of atoms was fantastic. He stubbornly worked on gravity for eight incredibly difficult years, despite being warned by Max Planck not to take on such an impossible problem, and in the end he came up with the magnificent general theory of relativity. And for almost 20 years he stood virtually alone in saying that light should be thought of as having particle properties, and finally in 1924 everyone else had to admit that they had been wrong and he had been right. In short, he had a pretty good track record by 1925. Why shouldn't his instincts turn out to be correct yet again?
In seeking to understand light's dual identity -- wave and particle -- he encouraged the concept of "matter waves" or "de Broglie waves." We've established that light is weird, but in 1924 a French doctoral student by the name of Louis de Broglie suggested that Nature might be weird but symmetric. That is, if things that were traditionally thought of as waves were also particles, wouldn't it be aesthetically pleasing if things that were traditionally thought of as particles were also waves? In other words, might electrons, atoms, and bowling balls have wave properties?
For reasons I won't go into, de Broglie proposed the following relationship between the particle model and the wave model:
Particles with large masses and large speeds correspond to short wavelength waves.
This is why we never see bowling balls acting like waves and diffracting around bowling pins: Their wavelengths are so ridiculously tiny that any slit or obstacle will be huge by comparison. Diffraction of any wave is negligible when the wavelength is much smaller than the size of the opening or obstacle.
(Remember, the wavelength of a bowling ball -- a wave property -- has nothing whatsoever to do with the ball's diameter -- a particle property. In order to picture the ball's wavelength, you first have to think of it as something that's oscillating and spread throughout the bowling alley rather than resting in your hands. If you find that difficult to visualize, join the club.)
Electrons, on the other hand, are the least massive particles known (other than zero-mass particles like photons), so their "de Broglie wavelengths" are small but not impossibly small. By 1927 electrons had been bounced off of crystals -- whose regular arrangement of atoms make them sort of like complicated 3-D diffraction gratings -- and had produced diffraction patterns on a phosphorescent screen. (Remember, traditional televisions and computer screens glow due to electrons striking a phosphor coating.) It was clear that de Broglie was right.
(I note in passing that this technique is used today in studying crystals, molecules, viruses, and so on. In lab, a widely spaced double-slit interference pattern told you that the slits were close together. Similarly, the diffraction pattern produced by X-rays, electrons, or neutrons scattered by a crystal tells you something about the arrangement of atoms within the crystal. We also rely on de Broglie's ideas every time we operate an electron microscope, a widely used device that uses "electron waves" rather than light waves to study small objects.)
By the late 1950s it was technologically possible to manufacture tiny double-slit patterns, send electron beams through the slits, magnify the resulting pattern, and verify that electrons behave just like waves. The weird part is that they also behave like particles: they have mass; they have charge; and they make a little "blip" on a phosphorescent screen when they strike it. That last bit is the really hard part to imagine, because a wave is a spread-out pattern -- and a spread-out thing can't make a flash of light at just one tiny place on a screen. If you try to imagine a water wave that makes the seaweed rise and fall at just one tiny spot along the coast, you'll start to see the problem.
Don't forget: All particles have wave properties. It's just easiest to observe this with electrons.
Werner Heisenberg's uncertainty principle states that we cannot simultaneously know, to arbitrary precision, both where something is and where it is going. If we precisely measure an object's location, we have no idea what a subsequent measurement of velocity (speed + direction) will give us. A highly precise velocity measurement, on the other hand, causes the object's probability cloud to "spread out," so that we have no idea where the object will show up if we now measure its position.
This is in accord with our ideas on wave/particle duality, since all "particles" (electrons, molecules, armchairs) have wave properties, and waves don't have definite locations. The term "uncertainty" is misleading, since it implies that objects have definite positions and velocities, but that we just can't know what they are. Sending electrons one at a time through a double slit demolishes that idea. If electrons had well-defined but unknown trajectories, we still could be sure that each one would go through either one slit or the other, but not both; yet when we actually do the experiment, the electrons slowly build up an interference pattern, showing that in some sense each one goes through both slits. Thus objects really don't have definite positions and velocities, a situation sometimes referred to as "quantum indeterminacy" rather than "quantum uncertainty."
It's important to note that the "uncertain" quantities aren't actually position and velocity, but position and momentum, where momentum equals mass times velocity. This is important because ordinary objects (like bowling balls) have very large masses and very large momenta. Hence the tiny quantum imprecision in their momentum is of no practical consequence. Electrons, on the other hand, have tiny masses and tiny momenta -- so quantum indeterminacy is a major factor in their behavior.
Quantum indeterminacy shows us why the Bohr model can't be right: electrons can't have a definite (circular) trajectory if they can't have a definite position and velocity at any one time. It also shows us why we don't get a double-slit interference pattern if we monitor which slit each electron goes through. By measuring the location of an electron well enough to know which slit it went through, we unavoidably "mess up" its velocity to the point that it might be heading in any number of directions. This in turn messes up our nice double-slit pattern.
Heisenberg came up with an (at the time) odd way of computing quantum outcomes -- say, an electron in an atom's third energy level jumping downward to the second level -- using matrices to do the math in a way that paid no attention at all to what actually happened to bring about a given outcome. There was no cause and effect, no process unfolding in space and time, just a starting condition and an outcome with no satisfying link between the two.
Erwin Schrödinger didn't have much use for this approach, so he came up with something more to the liking of most physicists: he described the behavior of electrons (and other particles) by means of a mathematical wave function, computed by means of a hideous expression today known as the Schrödinger equation. Both the equation and the resulting wave functions involve complex numbers -- that is, numbers that include the square root of -1. The good part was that Schrödinger hoped to have found an equation that would restore cause and effect, that would show how an electron-wave's behavior would depend on its environment, just as (for example) there are wave equations that show how a water wave's motion depends on the depth of the water it passes through. One problem, though, was that measurable physical quantities never involve the square root of -1. Even worse, Schrödinger eventually was able to show that his equation and Heisenberg's matrices, despite looking nothing at all like each other, were in fact mathematically equivalent ways of doing things: They would always yield the same numerical answers to any question. He hadn't done away with "this damned quantum jumping" after all. What kind of wave did this wave function describe?
In 1927 Max Born answered that question: The wave function describes probabilities. Suffice it to say that this function represents a "waving" of a complex sort, and that we can manipulate this function (by means of a sort of complex squaring operation) to obtain the probability of any quantum outcome we might choose to consider. In fact, we often refer to a pictorial representation of the squared wave function as a "probability cloud." (Chemists generally call it an "orbital.")
If you want to know the probability that a neutron will be found in some location, or that an electron is moving at some speed, or that a sodium atom will emit a yellow photon, or that a carbon-14 nucleus will undergo radioactive decay, just use Schrödinger's equation to compute the wave function for that object and then square it.
The uncertainty principle can be stated in terms of the wave function. For example, if an electron has a well known position, the wave function will give you little idea of what the velocity is: a wide range of velocities will have roughly equal probabilities. If you now precisely measure the velocity, the wave function changes at that instant. The new wave function tells you that the electron's velocity is almost certain to be whatever you just measured it to be. But it no longer gives you much of an idea of where the electron is: the probability cloud has spread out. You can only "ask" about one aspect of the electron at a time, you can't know everything at once -- a concept Bohr referred to as "complementarity."
(By the way, the jargon for this sudden change is "state reduction" or "collapse of the wave function." In class, we've sometimes simply said that the electron "decides" what to do at this instant, or that Nature "rolls the dice.")
The most infamous wave function of all is that of "Schrödinger's cat"; it is the sum of two mathematical terms, one of which describes a live cat and one of which describes a dead cat. Now, if only we understood what such a sum actually means, we'd be set. Schrodinger's point, of course, is that we don't really know what it means for electrons, or for neutrons, or for radioactive carbon-14 nuclei, and so on. This, he felt, shows us that this quantum stuff may produce useful answers, but if it implies that a cat could be both alive and dead simultaneously then it must be wrong. Einstein and de Broglie agreed with him, but few other physicists did (or do).
At what point do the radioactive nuclei in the box "decide" whether or not they have decayed, so that the cat can "decide" whether or not it has died? Is it true that the act of swinging a macroscopic hammer causes the nucleus to have definitely decayed, so that the cat is entirely described by a "dead" wave function? Is it instead true that the cat really is both dead and alive until human consciousness -- some person opening the box and looking at the cat -- stimulates a decision? Or is the cat conscious enough to ponder the fact that it is still alive, and thus collapse its own wave function so that it describes a living cat? Stranger still, do the two halves of the wave function correspond to two "parallel universes," one with a dead feline and one with a live one? All of these suggestions (and many more exist!) have their complications or drawbacks, and you should be somewhat familiar with these.
Einstein came up with his own thought experiment to demonstrate the absurdity (in his mind) of quantum mechanics. He published this along with Boris Podolsky and Nathan Rosen, so it's become known as the EPR experiment. It has to do with two "entangled" particles -- let's call them A and B -- that are in a superposition of quantum states, meaning that their properties (according to Bohr and company) aren't defined until we measure them, at which point Nature "flips a coin" (collapses the wave function) to choose an outcome. But the added ingredient is that this wave function guarantees that the states of A and B are perfectly correlated with each other, so that measuring A immediately tells us what we'll get when we measure B. One example is a pair of electrons that are guaranteed to have opposite spin directions, except that you don't know which of the two is spin-up and which is spin-down until you actually measure the spin for one of them. The wave function of these two electrons taken together represents a superposition of the two possibilities: (A up and B down) + (A down and B up). The example we considered in class is a pair of photons -- created when a calcium atom makes a rapid pair of downward quantum jumps -- that are guaranteed to have the same polarization. For this situation the wave function is the superposition (A vertical and B vertical) + (A horizontal and B horizontal), but until we measure one of the two we don't know whether we'll find that both are vertically polarized or instead that both are horizontally polarized.
Let's go with the two-photon example. If we send A and B in opposite directions and eventually measure A, finding (for example) that A is vertically polarized, then we know that B also must be measured to be vertically polarized, no matter how soon we measure it after the measurement on A. So a measurement of photon A must instantaneously influence photon B -- B must "know" how the random coin toss came out for A and must be sure to behave in the same way when measured -- even if B and A are many light-years apart by this time, in each other's absolute elsewhere. Einstein called this "spooky action at a distance" and claimed that reality couldn't be this strange, that B's polarization must have been definitely vertical from the time the two photons parted company, that B's polarization couldn't have been instantly influenced by a faraway measurement made on A. This idea is called "separability": it's possible for two objects A and B to be separated by such a great distance that a measurement made on the one can't possibly influence a measurement made on the other. Einstein's common sense view was that both separability and realism (the idea that a real world exists independent of our measurements) are valid concepts; this combined notion is called "local realism." Bohr, naturally, disagreed, because quantum mechanics denies the validity of both separability and realism.
Today the kind of theory espoused by Einstein is referred to as a "hidden variables" theory: photon B's polarization state is determined by some unmeasured property intrinsic to B, not by a distant measurement made on photon A. Again, this is common sense. If we measure identical twin A to have type-O blood, we can be sure that twin B will also have type-O blood; but we don't claim that B's blood type was undefined until we measured A's blood type! No, we recognize that B's blood type was type-O ever since B was born, due to B's genetic makeup; those genes, the unmeasured DNA sequence in B's chromosomes, are the "hidden variable" that determined in advance what we would measure for B, whether or not we'd chosen to measure A first.
In 1964 John Bell showed, to everyone's surprise, that one could actually test Einstein's idea in the lab, using a setup only somewhat more complicated than the one in the EPR thought experiment. He proved ("Bell's theorem") that if you want to predict correlations between the measured properties of entangled systems such as our two-photon system, hidden variable theories necessarily yield predictions that differ from the predictions of quantum mechanics. That is, if separability holds true in Nature (hidden variables) then the measured correlations must obey certain numerical constraints called "Bell inequalities," whereas if it doesn't hold true (quantum mechanics) then the lab data will violate those inequalities. By the early 1980s Alain Aspect and his collaborators had been able to carry out such an experiment, even taking care to make rapid changes in the angles of the two measuring devices so that they couldn't communicate with each other and "conspire" to throw the results. Einstein's intuition was wrong, it turns out: The experimental data clearly violated the Bell inequality, instead coming out exactly as quantum mechanics predicted. Nature doesn't respect separability; spooky action at a distance is How Things Are.
We can turn this weirdness to our advantage in quantum computing. I'll leave it to you to read up on the basic concepts here. (Optionally, if you want to have even more mind-wrenching fun with entanglement, get online or go to the library and read about quantum teleportation -- or just ask Vicki and Dave about it.)
All of this, however, came long after Einstein's 1955 death. His final years were devoted to his unsuccessful search for a "unification theory" that would merge electromagnetism with general relativity while doing away with quantum weirdness; he also devoted a lot of time to pacifism and Zionism. Today physicists are following in his footsteps, but realize that first one should unify electromagnetism with the strong and weak nuclear forces (a "grand unified theory") before attempting to include the vastly weaker force of gravity (a "theory of everything").