Our Mathematical Universe

by Max Tegmark

  Two waves can pass through each other unaffected, like the circular waves in the water tank in Figure 7.6 (left); at any time, their effects simply add together. In some places, we see peaks of the two waves adding up to an even higher peak (so-called constructive interference), in others we see a peak from one wave canceling the trough from the other to leave the water completely undisturbed (so-called destructive interference). On the surface of the Sun (Figure 7.6, center), sound waves in the hot gas/plasma have been observed. If such a wave propagates all the way around the Sun (right), then it will cancel itself out with destructive interference unless it performs exactly a whole number of oscillations as it goes around, thereby staying in sync with itself. This means that, just as a flute, the Sun vibrates only with certain special frequencies.1 In his 1924 Ph.D. thesis, de Broglie applied this reasoning to waves going around the hydrogen atom instead of the Sun, and obtained the exact same frequencies and energies as the Bohr model had predicted. A more direct demonstration of particles behaving as waves is given by the double-slit experiment illustrated in Figure 7.7.

  Figure 7.6: Waves in a water tank (left) and on the Sun (right)

  This wave picture also gives a more intuitive picture of why atoms don’t collapse as classical physics predicted: if you try to confine a wave to a very small space, it immediately starts spreading out. For example, if a raindrop lands in a water tank, it will initially disturb the water only in the small area where it landed, but this disturbance soon starts spreading outward in all directions as a series of circular waves, like the ripples in Figure 7.6 (left). This is the essence of the Heisenberg uncertainty principle: Werner Heisenberg showed that if you confine something to a small region of space, then it will have lots of random momentum, which tends to make it spread out and become less confined. In other words, an object can’t simultaneously have an exact position and an exact velocity!2 This means that if a hydrogen atom tries to collapse as in Figure 7.5 (left) by sucking the electron into the proton, then the increasingly confined electron will get enough momentum and speed to come flying back out to a higher orbit again.

  Figure 7.7: If we fire particles (say, electrons or photons from a laser gun) at a barrier with two vertical slits, classical physics predicts that they’ll hit our detector in two vertical strips behind the slits. In contrast, quantum mechanics predicts that each particle will act like a wave and pass through both slits in a quantum superposition, interfere with itself, and form an interference pattern akin to Figure 7.6. Performing this famous double-slit experiment shows that quantum mechanics is correct: one detects particles along a whole series of vertical strips.

  De Broglie’s thesis made waves, and in November 1925, Erwin Schrödinger gave a seminar about it in Zurich. When he was finished, Peter Debye said in effect: “You speak about waves, but where is the wave equation?” Schrödinger went on to produce and publish his famous wave equation (Figure 7.4), the master key for so much of modern physics. An equivalent formulation involving tables of numbers called matrices was provided by Max Born, Pasqual Jordan and Werner Heisenberg around the same time. With this new powerful mathematical underpinning, quantum theory made explosive progress. Within a few years, a host of hitherto unexplained measurements had been successfully explained, including spectra of more complicated atoms and various numbers describing properties of chemical reactions. Eventually this quantum physics gave us the laser, the transistor, the integrated circuit, computers and smartphones. Further successes of quantum mechanics involve its extension, quantum field theory, which underpins present-day frontier research such as the search for dark-matter particles.

  What’s the hallmark of good science? There are several science definitions that I like, and one of them is data compression, explaining a lot with a little. With a good scientific theory, you get more out of it than you put into it. I just applied standard data-compression software to the text file containing this chapter draft, and it compressed it threefold, using regularities and patterns that it found in my prose. Let’s compare this to quantum mechanics. I just downloaded a list of over 20,000 spectral lines from http://physics.nist.gov/cgi-bin/ASD/lines1.pl that have had their frequency painstakingly measured in laboratories around the world, and by capturing the patterns and regularities in these numbers, the Schrödinger equation can data-compress them down to just three numbers: the so-called fine-structure constant α ≈ 1/137.036, which gives the strength of electromagnetism, the number 1836.15, which is how many times heavier the proton is than the electron, and the orbital frequency of hydrogen.3 That’s the equivalent of data compressing this whole book down to a single sentence!

  Erwin Schrödinger is one of my physics superheros. When I was a postdoc at the Max Planck Institute for Physics in Munich, the copying machine in the library used to take eons to warm up, and I’d pass the time by pulling classic books from the shelves. Once I pulled out Annalen der Physik from 1926, and was amazed to see that essentially everything we’d covered in my graduate quantum classes had been worked out in four of his 1926 papers. I admire him because he wasn’t just brilliant, but also a freethinker: he questioned authority, thought for himself and did what he felt was right. After getting Max Planck’s job as professor in Berlin, one of the most prestigious posts in the world, he gave it up because he wouldn’t tolerate Nazi persecution against his Jewish colleagues. He then turned down a job offer from Princeton because they wouldn’t accept his unorthodox family decisions (he lived with two women and had a child with the one he wasn’t married to). Indeed, when I made a pilgrimage to his grave during a 1996 ski vacation in Austria, I discovered that his freethinking didn’t go down well in his home village either: you’ll see in the photo I took (Figure 7.4) that the small town of Alpbach has buried their most famous citizen ever in a quite modest grave right at the edge of the cemetery.…

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  1The same phenomenon has been observed in car tires at very high speeds, and the resulting sound waves traveling around the tube in resonance can damage your budget.

  2Specifically, if a particle has position uncertainty Δx and momentum uncertainty Δp, then the Heisenberg uncertainty principle states that Δx Δp ≥ ħ/2 where ħ is the reduced Planck constant h/2π as before. Mathematically, the uncertainty for each quantity is defined as the standard deviation of its probability distribution.

  3In fact, the last one arguably shouldn’t count because we could redefine our unit of time so that it equals 1. If you want still more accuracy to get all the measured decimal places right, you need only toss in a few more numbers to better model the exact masses of the different atomic nuclei (neutrons weigh about 0.1% more than protons, and so on).

  Quantum Weirdness

  But what did it all mean? What were these waves that Schrödinger’s equation described? This central puzzle of quantum mechanics remains a potent and controversial issue to this day.

  When we physicists describe something mathematically, we usually need to describe two separate things:

  1. Its state at a given time.

  2. The equation describing how this state will change over time.

  For example, to describe the orbit of Mercury around the Sun, Newton described the state of Mercury by six numbers: three for the position of its center (say, the x-, y- and z-coordinates) and three for the components of the velocity in these directions.1 For the equation of motion, he used Newton’s law: that the acceleration is given by the gravitational pull toward the Sun, which depends on the inverse square of the distance to the Sun.

  In his solar-system atom model (Figure 7.5, middle), Niels Bohr changed the second part of the description by introducing quantum jumps between special orbits, but he kept the first part. Schrödinger was even more radical, and changed the first part too: he abandoned the very idea that a particle has a well-defined position and velocity! Instead, he described the state of a particle by a new mathematical beast called a wavefunction, written ψ, which describes the extent to which the particle is in diff
erent places. Figure 7.5 (right) shows the square2 of the wavefunction, |ψ|2, for the electron in a hydrogen atom in an n = 3 orbit, and you can see that rather than being in a particular place, it seems to be on all sides of the proton equally, while preferring certain radii over others. How intense the “electron cloud” of Figure 7.5 (right) is in different places corresponds to the extent to which the electron is in these places. Specifically, if you experimentally go looking for the electron, you find that the square of the wavefunction gives the probability that you’ll find it in different places, so some physicists like to think of the wavefunction as describing a probability cloud or probability wave. In particular, you’ll never find a particle in places where its wavefunction equals zero. If you want to stir up a cocktail party by sounding like a quantum physicist, another buzzword you’ll need to drop is superposition: a particle that’s both here and there at once is said to be in a superposition of here and there, and its wavefunction describes all there is to know about this superposition.

  Figure 7.8: The wavefunction ψ on the verge of collapse

  These quantum waves are strikingly different from the classical waves from Figure 7.6: a classical wave that you’re surfing on is made of water and the thing which has a wavy shape is the water surface, but the thing which is wavy or cloudlike in a hydrogen atom isn’t water or any kind of substance at all: there’s only a single electron there, and what’s wavy is its wavefunction, the extent to which it is in different places.

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  1If vector calculus floats your boat, then think of the state as simply the position vector r and its time derivative r⋅ (the velocity vector).

  2If you’re a math aficionado and like complex numbers, you’ll be pleased to know that the wavefunction for a particle specifies a complex number ψ (r) for each place r in space. What I’m casually calling the “square” of the wavefunction throughout this book for brevity is actually |ψ|2, the square of the absolute value |ψ| of the wavefunction, which is defined as the real part squared plus the imaginary part squared. If you’re not a math aficionado, then don’t worry, since you can understand the key arguments in this book anyway.

  The Collapse of Consensus

  In summary, Schrödinger altered the classical description of the world in two ways:

  1. The state is described not by positions and velocities of the particles, but by a wavefunction.

  2. The change of this state over time is described not by Newton’s or Einstein’s laws, but by the Schrödinger equation.

  These discoveries by Schrödinger have been universally celebrated as among the most important achievements of the twentieth century, and they created a revolution in both physics and chemistry. But they also left people tearing their hair out in confusion: if things could be in several places at once, why did we never observe that (while sober)? This puzzle became known as the measurement problem (in physics, measurement and observation are synonyms).

  After much debate and discussion, Bohr and Heisenberg came up with a remarkably radical remedy that became known as the Copenhagen interpretation, which to this day is taught and advocated in most quantum-mechanics textbooks. A key part of it is to add a loophole to the second item mentioned above, postulating that change is only governed by the Schrödinger equation part of the time, depending on whether an observation is taking place. Specifically, if something is not being observed, then its wavefunction changes according to the Schrödinger equation, but if it is being observed, then its wavefunction collapses so that you find the object only in one place. This collapse process is both abrupt and fundamentally random, and the probability that you find the particle in any particular place is given by the square of the wavefunction. The wavefunction collapse thus conveniently gets rid of schizophrenic superpositions and explains our familiar classical world where we see things in only one place at a time. Table 7.3 summarizes the key quantum concepts that we’ve explored so far, and how they’re interrelated.

  There are other elements to the Copenhagen interpretation as well, but the part above is what’s most agreed on. I’ve gradually discovered that those of my colleagues who hail Copenhagen as their favorite interpretation of quantum mechanics usually disagree with each other about some of those other elements, making it more appropriate to talk about the “Copenhagen interpretations.” The relativity pioneer Roger Penrose quipped: “There are probably more different attitudes to quantum mechanics than there are quantum physicists. This is not inconsistent because certain quantum physicists hold different views at the same time.” Indeed, even Bohr and Heisenberg held slightly different views about what it implied about the nature of reality. However, all physicists back then agreed that the Copenhagen interpretation worked great for simply getting on with business as usual in the lab.

  Not everyone was thrilled, however. If wavefunction collapse really happened, then this would mean that a fundamental randomness was built into the laws of nature. Einstein was deeply unhappy about this interpretation, and expressed his preference for a deterministic universe with the oft-quoted remark “I can’t believe that God plays dice.” After all, the very essence of physics had been to predict the future from the present, and now this was supposedly impossible not just in practice, but even in principle. Even if you were infinitely wise and knew the wavefunction of the entire Universe, you couldn’t calculate what the wavefunction would be in the future, because as soon as someone in our Universe made an observation, the wavefunction changed randomly.

  Quantum-Mechanics Cheat Sheet

  Wavefunction Mathematical entity describing the quantum state of an object. The wavefunction of a particle describes the extent to which it’s in different places

  Superposition Quantum-mechanical situation where something is in more than one state at once, for example in two different places

  Schrödinger equation Equation that lets us predict how the wavefunction will change in the future

  Hilbert space Abstract mathematical space where the wavefunction lives

  Wavefunction collapse Hypothesized random process whereby the wavefunction changes abruptly in violation of the Schrödinger equation, giving a measurement a definite outcome. Lack of wavefunction collapse implies Hugh Everett’s Level III multiverse

  Measurement problem The controversial question of what happens to the wavefunction during a quantum measurement: does it collapse or not?

  Copenhagen interpretation A set of assumptions including that the wavefunction collapses during measurements

  Everett interpretation The assumption that the wavefunction never collapses—implies the Level III multiverse (Chapter 8)

  Decoherence A censorship effect derivable from the Schrödinger equation, whereby superpositions become unobservable unless they’re kept secret from the rest of the world—makes the wavefunction appear to collapse during measurements even if it actually doesn’t (Chapter 8)

  Quantum immortality The idea that we’re subjectively eternal if the Level III multiverse exists. I suspect that there’s no quantum immortality because the continuum is an illusion (Chapter 11).

  Table 7.3: Summary of key quantum-mechanics concepts (Hilbert space and the last three concepts will be introduced in the next chapter)

  Another aspect of collapse that caused consternation was that observation was upgraded to such a central concept. When Bohr exclaimed, “No reality without observation!” it seemed to put humans back on center stage. After Copernicus, Darwin and others had gradually deflated our human hubris and warned against our egocentric tendencies to assume that everything revolved around us, the Copenhagen interpretation made it seem as if we humans in some sense created reality by just looking at it.

  Finally, some physicists were irked by the lack of mathematical rigor. Whereas traditional physical processes would be described by mathematical equations, the Copenhagen interpretation had no equation specifying what constituted an observation, that is, exactly when the wavefunction would collapse. Did it really require a human
observer, or was consciousness in some broader sense sufficient to collapse the wavefunction? As Einstein put it: “Does the Moon exist because a mouse looks at it?” Can a robot collapse the wavefunction? What about a webcam?

  The Weirdness Can’t Be Confined

  Loosely speaking, the Copenhagen interpretation of quantum mechanics suggests that small things act weird but big things don’t. Specifically, things as small as atoms are usually in several places at once, but things as big as people aren’t. The above-mentioned gripes aside, this is a tenable view as long as the weirdness stays confined to the microworld and doesn’t somehow leak into the macroworld, like an evil genie being confined to a bottle, unable to get big and wreak havoc. But does it really stay confined?

  One of the things that bothered me back in that Stockholm dorm room at the beginning of this chapter was that big things are made of atoms, and so since atoms can be in several places at once, they can be, too. But just because they can doesn’t mean that they will: you might hope that there are no physical processes that amplify microscopic weirdness into macroscopic weirdness. Schrödinger himself shattered such hopes with a diabolical thought experiment: Schrödinger’s cat is trapped in a box with a cyanide canister that’s opened if a single radioactive atom decays. After a while, the atom will be in a superposition of decayed and not decayed, which causes the entire cat to be in a superposition of dead and alive. In other words, a seemingly innocent microsuperposition involving a single atom is amplified over time into a macrosuperposition where a cat containing octillions of particles is in two states at once. Moreover, such weirdness amplification happens all the time, even without sadistic contraptions. You may have heard about chaos theory: how the laws of classical physics can exponentially amplify tiny differences, such as a Beijing butterfly perturbing the air and ultimately causing a Stockholm storm. An even simpler example is a pencil balanced on its tip, where a microscopic nudge of the initial tilt can determine the direction in which it will ultimately come crashing down. Whenever such chaotic dynamics are at play, the initial position of a single atom can make all the difference, so if that atom is in two places at once, you’ll end up with macroscopic things in two places at once.