by Max Tegmark
Table 7.1: All known elementary particles are described by their own unique sets of quantum numbers, and this table shows a sample. The particles are purely mathematical objects in the sense that they have no properties at all beyond their quantum numbers. The mass shown corresponds to how much energy you’d need to create the particle at rest. The funny unit MeV is the amount of motion energy an electron picks up if you use a million volts to accelerate it.
Click here to see a larger image.
I remember this old Cold War joke about how, in the West, everything that wasn’t forbidden was allowed, while in the East, everything that wasn’t allowed was forbidden. Intriguingly, particle physics seems to prefer the former: every reaction that isn’t forbidden (for violating some conservation law) appears to actually occur in nature. This means that we can think of the fundamental Legos of particle physics as being not the particles themselves, but the conserved quantities! So particle physics is simply rearranging energy, momentum, charge and other conserved quantities in new ways. For example, Table 7.1 shows that the cookbook recipe for making an up quark is to combine 2/3 units of charge, 1/2 unit of spin, 1/2 unit of isospin, 1/3 unit of baryon number, and top it all off with a few MeV of energy.
So what are quantum numbers like energy and charge made of? Nothing—they’re just numbers! A cat has energy and charge, too, but it also has many other properties besides these numbers such as its name, smell and personality—so it would sound crazy to say that the cat is a purely mathematical object completely described by those two numbers. Our elementary-particle friends, on the other hand, are completely described by their quantum numbers, and appear to have no intrinsic properties at all besides these numbers! In this sense, we’ve now come full circle back to Plato’s idea: the fundamental Legos out of which everything is made appear to be purely mathematical in nature, having no properties except mathematical properties. We’ll return to this idea in more detail in Chapter 10, and see that it’s just the tip of a mathematical iceberg.
At a more technical level, some particle physicists like to glibly answer the question “What’s a particle?” by saying, “It’s an element of an irreducible representation of the symmetry group of the Lagrangian.” That’s quite a mouthful, and enough to stop most budding conversations dead in their tracks, but it’s a completely mathematical thing, just a bit more general than the concept of a set of numbers. And yes, sure, string theory or a competitor may deepen our understanding of what particles really are, but all the leading theories out there simply replace one mathematical entity with another. For example, if the quantum numbers from Table 7.1 turn out to correspond to different types of superstring vibrations, then you shouldn’t think of these strings as fuzzy little objects with intrinsic properties like being made out of braided golden-brown cat hairs, but rather as purely mathematical constructs that physicists have dubbed “strings” simply to emphasize their one-dimensional nature and to make an analogy with something that feels less mathematical and more familiar.
In summary, nature has a hierarchical Lego structure. If my son Alexander plays normally with his birthday present, all he can rearrange are the factory-made Lego pieces. If he’d play atom Lego by setting them on fire, immersing them in acid, or using some alternative method to rearrange their atoms, he’d be doing chemistry. If he’d play nucleon Legos by rearranging their neutrons and protons into different kinds of atoms, he’d be doing nuclear physics. If he’d smash his pieces together near the speed of light to rearrange the energy, momentum, charge, etc., of their neutrons, protons and electrons into new particles, he’d be doing particle physics. The Legos at the deepest level appear to be purely mathematical objects.
Particle-Physics Cheat Sheet
Momentum The punch something packs if it crashes into something or, more rigorously, the amount of time it would take you to stop it times the average force with which you’d need to push it
Angular momentum How much something spins or, more rigorously, the amount of time it would take you to make it stop spinning times the average torque (twisting force) you’d need to use
Spin The angular momentum of a single particle spinning around its center
Conserved quantity Quantity that remains constant over time and can neither be created nor destroyed. Examples: energy, momentum, angular momentum, electric charge
Atom Electrons orbiting around a nucleus of protons and neutrons; the number of protons in an atom determines its name (1 = hydrogen, 2 = helium, etc.)
Electron Negatively charged particle that electric currents are made of
Proton Positively charged particle found in atomic nuclei, made of two up quarks and a down quark
Neutron Particle without electric charge that’s found in atomic nuclei, made of two down quarks and an up quark
Photon Particle of light
Gluon Particle that help glue quark triplets together into protons and neutrons
Neutrino Particle that’s so stealthy that it can usually pass right through Earth without interacting with anything
Fermion Particle that can’t be in the same place and state as an identical particle. Examples: electrons, quarks, neutrinos
Boson Particle that likes to be in the same place and state as an identical particle. Examples: photons, gluons, Higgs particle
Table 7.2: Summary of key physics terms for understanding the microworld
* * *
1The momentum of something measures the punch it packs if it crashes into something; or, more rigorously, the amount of time it would take you to stop it times the average force with which you’d need to push it. The momentum p of something with mass m moving with velocity v is simply given by p = mv as long as v is far below the speed of light.
Photon Legos
It’s not only “stuff” that’s made of Lego-like building blocks. As we mentioned in Part I of this book, so is light, being composed of particles called photons, inferred by Einstein in 1905.
Four decades earlier, James Clerk Maxwell had discovered that light is an electromagnetic wave, a type of electrical disturbance. If you could carefully measure the voltage between two points in a beam of light, you’d find that it oscillates over time; the frequency f of this oscillation (how many times per second it oscillates) determines the color of the light, and the strength of the oscillation (the maximum number of volts you measure) determines the intensity of the light. Our Omniscope, from back in Chapter 4, measures such voltages. We humans give these electromagnetic waves different names, depending on their frequency (by increasing frequency, we call them radio waves, microwaves, infrared, red, orange, yellow, green, blue, violet, ultraviolet, x-rays, gamma rays), but they’re all forms of light and they’re all made of photons. The more photons an object emits each second, the brighter it looks.
Einstein realized that the amount of energy E in a photon was given by its frequency f through the simple formula E = hf, where h is the constant of nature known as Planck’s constant. The constant h is tiny, so a typical photon has very little energy in it. If I lie on the beach for just a second, I get warmed by about a sextillion (1021) photons, which is why it feels like a continuous flow of light. However, if my friends have sunglasses blocking 90% of the light, and I put on twenty-one pairs at once, then only about one of the original photons would get through each second, which a sensitive photon detector could confirm.
Einstein got the Nobel Prize because he used this idea to explain the so-called photoelectric effect, whereby the ability of light to knock electrons out of metal had been found to depend only on the frequency of the light (the energy of the photons), not on the intensity (the number of photons). Lower-frequency photons just don’t have enough energy for the task, just as you can’t break a glass window by throwing tennis balls with low energy no matter how many you throw. The photoelectric effect is related to the processes used in present-day solar cells and the image sensors in digital cameras.
My namesake Max Planck won the 1918 Nobel Priz
e for showing that the photon idea also solved another outstanding mystery: why the previously calculated heat radiation of a glowing hot object didn’t come out right. The rainbow (Figure 2.5) reveals the spectrum of sunlight, that is, how much light there is at different frequencies. People knew that the temperature T of something is a measure of how rapidly its particles are moving around, and that the typical motion energy E of a particle was given by the formula E = kT, where k is a number known as Boltzmann’s constant. When particles in the Sun collide, roughly a quantity kT of motion energy can be converted into light energy. Unfortunately, the detailed prediction for the rainbow was the so-called ultraviolet catastrophe: that the intensity of light would increase forever toward the right in Figure 2.5 (toward higher frequencies), so that you’d get blinded by gamma rays when you looked at any warm object, say your best friend. You’re saved by the fact that light is made of particles: the Sun can radiate light energy only one photon at a time, and the typical energy kT available for making a photon falls far short of the amount of energy hf required to make even a single gamma ray.
Above the Law?
So if everything is made of particles, then what are the laws of physics that govern them? Specifically, if we know what all the particles in our Universe are doing right now, then what equation lets us calculate what they’ll be doing in the future? If there is such an equation, then you might hope that it will allow us to—at least in principle—predict all aspects of the future from the present, from the future trajectory of a just-hit baseball to the winners of the 2048 Olympic Games: just figure out what all the particles will do, and there’s your answer.
The good news is that there does seem to be just such an equation, called the Schrödinger equation (Figure 7.4). The bad news is that it doesn’t predict exactly what the particles will do, and that almost a century after the Austrian physicist Erwin Schrödinger wrote it down, physicists still argue about what to make of it.
What everybody does agree on is that microscopic particles don’t obey the classical laws of physics that we’re taught in school. Since an atom is reminiscent of a miniature solar system (Figure 7.1), it would seem quite natural to assume that its electrons orbit the nucleus according to Newton’s laws just as the planets orbit the Sun. Indeed, when you do the math, things look promising at first. You can spin a yo-yo in a circle around your head by pulling with a force on its string; if the string snapped, the yo-yo would move in a straight line with constant speed, so the force with which you pull on it is required to deflect it from this straight-line motion to go in a circle. In our Solar System, it’s not a string but the Sun’s gravity that provides this force, and in an atom, the electric attraction of the nucleus provides the force. If you do the calculation for an orbit the size of a hydrogen atom, you’ll predict that the electron orbits just about as fast as we measure in the lab—quite a theoretical triumph! However, to be more accurate, we need to include one more effect in the math: an electron that’s accelerating (changing its speed or its direction of motion) will radiate away energy—your mobile phone exploits this by jiggling electrons around in its antenna so that radio waves get transmitted. Since energy is conserved, this radiated energy has to come from somewhere. In your phone, it gets taken from the battery, but in a hydrogen atom, it gets taken from the motion energy of the electron, causing it to fall farther and farther “down” toward the atomic nucleus, just as upper atmosphere air resistance causes satellites in low-Earth orbit to lose motion energy and eventually fall down. This means that the electron orbit isn’t a circle, but a death spiral (Figure 7.5): after about 100,000 orbits, the electron has crashed into the proton and the hydrogen atom has collapsed, at the ripe old age of about 0.02 nanoseconds.1
Figure 7.4: The Schrödinger equation lives on. Since I took this photo in 1996, the inscription font has mysteriously changed. Will quantum weirdness never end?
Click here to see a larger image.
This is bad. Really bad. Here, we’re not talking about some minor 1% discrepancy between theory and experiment, but about a prediction that all hydrogen atoms (as well as all other atoms) in our Universe will collapse in a billionth of the time it took you to read the last word in this sentence. Indeed, since most hydrogen atoms have been around for about 14 billion years, they’ve lasted more than twenty-eight orders of magnitude longer than classical physics predicts—this held the dubious record as the worst-ever quantitative failure of physics until it was overtaken by the 123-order-of-magnitude mismatch between prediction and measurement for the dark-energy density that we mentioned in Chapter 3.
When physicists assumed that elementary particles obeyed the classical laws of physics, they ran into other problems as well. For example, it was found that the amount of energy needed to heat very cool objects was smaller than predicted. There were also further problems, but we don’t need to flog a dead horse, since the message from nature is crystal clear: microscopic particles violate the laws of classical physics.
So are microscopic particles above the law? No, they obey a different law: Schrödinger’s.
* * *
1The electron makes about 1/8πα3 ~ 105 orbits before crashing into the proton, where α ≈ 1/137.03599968 is the dimensionless strength of the electromagnetic force, a.k.a. the fine-structure constant. You’ll find a nice derivation of the death spiral here: http://www.physics.princeton.edu/~mcdonald/examples/orbitdecay.pdf.
Quanta and Rainbows
To explain how atoms worked, the Danish physicist Niels Bohr introduced a radical idea in 1913: perhaps it wasn’t just matter and light that was quantized (that came in Lego-like discrete chunks), but aspects of motion as well. What if motion isn’t continuous but jumpy as in the computer game PAC-MAN or in an old Chaplin movie where the frame rate is too slow? Figure 7.5 shows Bohr’s atom model: circular orbits are allowed only if the circles have certain magical sizes. There’s a smallest allowed orbit labeled n = 1, and then there are larger orbits labeled n = 2, etc., the radii of which are n2 times as large as the smallest one.1
Figure 7.5: Our evolving understanding of the hydrogen atom. The classic “solar system” model of Ernest Rutherford was unfortunately unstable, with the electron spiraling into the proton in the center (I’m showing what it would look like if the electric force were twenty times stronger; otherwise it would spiral around about a hundred thousand times and make my plot illegible). The Bohr model confines the electron to discrete orbits numbered n = 1, 2, 3, etc., between which it can jump when emitting or absorbing photons, but fails for all atoms except hydrogen. The Schrödinger model has a single electron in many places at once, in an “electron cloud” whose shape is given by a so-called wavefunction ψ.
The first and most obvious success is that Bohr’s atom can’t collapse like the classical one to its left in Figure 7.5; when the electron is in the innermost orbit, there’s simply no smaller orbit for it to jump to. But Bohr’s model explains much more. The higher orbits have more energy than the lower ones, and total energy is conserved, so whenever the electron makes a PAC-MAN-like jump down to a lower orbit, the extra energy must get emitted from the atom in the form of a photon (see Figure 7.5), and in order to jump back up to a higher orbit, the electron must be able to pay the energy cost by absorbing an incoming photon with the required energy. Since there’s only a discrete set of orbit energies, this means that the atom can only emit or absorb photons with certain magical energies. In other words, an atom can only emit or absorb light of certain special frequencies. This solved a long-standing mystery: the rainbow of sunlight (Figure 2.5) was known to have dark lines at certain mysterious frequencies (certain colors were missing), and by studying hot glowing gases in the laboratory, it had been observed that each type of atom had its unique spectral fingerprint in the form of the frequencies of light that it could emit and absorb. Bohr’s atom model didn’t just explain the existence of these spectral lines, but also their exact frequencies for hydrogen.2
That was the good news, f
or which Bohr won a Nobel Prize (as did most of the others I mention in this chapter). The bad news was that Bohr’s model didn’t work for any atoms other than hydrogen, except if all but one of their electrons were removed.
* * *
1Actually, what Bohr did, which was mathematically equivalent, was to assume that the angular momentum of the electron orbit was quantized, and was only allowed to equal some multiple n of what’s called the reduced Planck constant ħ, defined as h/2π. Analogously to momentum, you can think of the angular momentum of a spinning object as a measure of the amount of time it would take you to make it stop spinning times the average torque (twisting force) you’d need to use. Something orbiting in a circle of radius r with momentum p has angular momentum rp.
2The energies of the orbits are E1/n2 where E1 is the known energy of the lowest orbit, so by jumping between two orbits n1 and n2, the electron can emit photons of all energies of the form E1.
Making Waves
Despite these early successes, physicists still didn’t know what to make of these strange and seemingly ad hoc quantum rules. What did they really mean? Why is angular momentum quantized? Is there a deeper explanation for this?
Louis de Broglie proposed one: that the electron (and indeed all particles) has wavelike properties the way photons do. In a flute, standing sound waves can vibrate only at certain special frequencies, so could something analogous be determining the frequencies with which electrons could orbit in atoms?
