It’s been over a century since Max Planck came up with the idea that electromagnetic energy could only be emitted in fixed units called quanta as a means of explaining the observed spectrum of light from incandescent light bulbs. Starting from this point, great physicists such as Bohr, de Broglie, Schrödinger, and Dirac developed the field of quantum mechanics, revolutionizing our understanding of the physical universe. By the 1930’s it was known that matter, as well as electromagnetic energy, could be described by wave equations. In other words, at the level of the atom, particles do not behave at all as if they were billiard balls on a table, or, in general, in the way that our senses portray physical objects to us at a much larger scale. For example, electrons don’t act like hard little balls flying around outside the nuclei of atoms. Rather, it is necessary to describe where they are in terms of probability distributions, and how they act in terms of wave functions. It is impossible to tell at any moment exactly where they are, a fact formalized mathematically in Heisenberg’s famous Uncertainty Principle. All this has profound implications for the very nature of reality, most of which, even after the passage of many decades, are still unknown to the average lay person. Among other things, it follows from all this that there are two basic types of elementary particles; fermions and bosons. It turns out that they behave in profoundly different ways, and that the idiosyncrasies of neither of them can be understood in terms of classical physics.
Sometimes the correspondence between mathematics and physical reality seems almost magical. So it is with the math that predicts the existence of fermions and bosons. When it was discovered that particles at the atomic level actually behave as waves, a brilliant Austrian scientist named Erwin Schrödinger came up with a now-famous wave equation to describe the phenomenon. Derived from a few elementary assumptions based on some postulates derived by Einstein and others relating the wavelength and frequency of matter waves to physical quantities such as momentum and energy, and the behavior of waves in general, the Schrödinger equation could be solved to find wave functions. It was found that these wave functions were complex numbers, that is, they had a real component, and an “imaginary” component that was a multiple of i, the square root of minus one. For example, such a number might be written down mathematically as x + iy. Each such number has a complex conjugate, found by changing the sign of the complex term. The complex conjugate of the above number is, therefore, x – iy. Max born found that the probability of finding a physical particle at any given point in space and time could be derived from the product of a solution to Schrödinger’s equation and its complex conjugate.
So far, so good, but eventually it was realized that there was a problem with describing particles in this way that didn’t arise in classical physics; you couldn’t tell them apart! Elementary particles are, after all, indistinguishable. One electron, for example, resembles every other electron like so many peas in a pod. Suppose you could put two electrons in a glass box, and set them in motion bouncing off the walls. Assuming you had very good eyes, you wouldn’t have any trouble telling the two of them apart if they behaved like classical billiard balls. You would simply have to watch their trajectories as they bounced around in the box. However, they don’t behave like billiard balls. Their motion must be described by wave functions, and wave functions can overlap, making it impossible to tell which wave function belongs to which electron! Trying to measure where they are won’t help, because the wave functions are changed by the very act of measurement.
All this was problematic, because if elementary particles really were indistinguishable in that way, they also had to be indistinguishable in the mathematical equations that described their behavior. As noted above, it had been discovered that the physical attributes of a particle could be determined in terms of the product of a solution to Schrödinger’s equation and its complex conjugate. Assuming for the moment that the two electrons in the box didn’t collide or otherwise interact with each other, that implies that the solution for the two particle system would depend on the product of the solution for both particles and their complex conjugates. Unfortunately, the simple product didn’t work. If the particles were labeled and the labels switched around in the solution, the answer came out different. The particles were distinguishable! What to do?
Well, Schrödinger’s equation has a very useful mathematical property. It is linear. What that means in practical terms is that if the products of the wave functions for the two particle system is a solution, then any combination of the products will also be a solution. It was found that if the overall solution was expressed as the product of the two wave functions plus their product with the labels of the two particles interchanged, or of the product of the two wave functions minus their product with the labels interchanged, the resulting probability density function was not changed by changing around the labels. The particles remained indistinguishable!
The solution to the Schrödinger equation, referred to mathematically as an eigenfunction, is called symmetric in the plus case, and antisymmetric in the minus case. It turns out, however, that if you do the math, particles act in very different ways depending on whether the plus sign or the minus sign is used. And here’s where the magic comes in. So far with just been doing math, right? We’ve just been manipulating symbols to get the math to come out right. Well, as the great physicist, Richard Feynman, once put it, “To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature.” So it is in this case. The real particles act just as the math predicts, and in ways that are completely unexplainable in terms of classical physics! Particles that can be described by an antisymmetric eigenfunction are called fermions, and particles that can be described by an symmetric eigenfunction are called bosons.
How do they actually differ? Well, for reasons I won’t go into here, the so-called exclusion principle applies to fermions. There can never be more than one of them in exactly the same quantum state. Electrons are fermions, and that’s why they are arranged in different levels as they orbit the nucleus of an atom. Bosons behave differently, and in ways that can be quite spectacular. Assuming a collection of bosons can be cooled to a low enough temperature they will tend to all condense into the same low energy quantum state. As it happens, the helium atom is a boson. When it is cooled below a temperature of 2.18 degrees above absolute zero, it shows some very remarkable large scale quantum effects. Perhaps the weirdest of these is superfluidity. In this state, it behaves as if it had no viscosity at all, and can climb up the sides of a container and siphon itself out over the top!
No one really knows what matter is at a fundamental level, or why it exists at all. However, we do know enough about it to realize that our senses only tell us how it acts at the large scales that matter to most living creatures. They don’t tell us anything about its essence. It’s unfortunate that now, nearly a century after some of these wonderful discoveries about the quantum world were made, so few people know anything about them. It seems to me that knowing about them and the great scientist who made them adds a certain interest and richness to life. If nothing else, when physicists talk about the Higgs boson, it’s nice to have some clue what they’re talking about.
