Quantum Reality Read online

  Alternatively, we conclude that the wave nature of the electron is an intrinsic behaviour. Each individual electron behaves as a wave, described by a wavefunction, passing through both slits simultaneously and interfering with itself before striking the screen.

  So, how are we supposed to know precisely where the next electron will appear?

  The wavefunction gives us only probabilities: in quantum mechanics we can only know what might happen, not what will happen

  A wave alternates between positive amplitude, largest at a peak, and negative amplitude, largest at a trough. We calculate the intensity of the wave as the square of its amplitude, which is always a positive number. So, in two-slit interference described purely in terms of waves we imagine a resulting wave which, when squared, produces a pattern which alternates between regions of high intensity (bright fringes) and zero intensity (dark fringes), as shown in Figure 5a.

  Figure 5 (a) Before measurement, the square of the electron wavefunction predicts a distribution of probabilities for where the electron might be found, spread across the screen. (b) After measurement, the electron is recorded to be found in one, and only one, location on the screen.

  But, by its very nature, this pattern of intensity is spread across the screen. It is distributed through space, or delocalized. And yet we know that in the experiment with electrons, as illustrated in Figure 4, we see electrons detected one at a time, as single bright spots, in only one location on the screen. Each electron hitting the screen is localized.

  How does this work?

  Schrödinger had wanted to interpret the wavefunction literally, as the theoretical representation of a ‘matter wave’. He argued that atoms are simply the diffraction patterns of electron waves captured and wrapped around atomic nuclei. But to make sense of one-electron interference we must reach for an alternative interpretation suggested later in 1926 by Max Born.

  Born reasoned that in quantum mechanics the square of the wavefunction is a measure not of the intensity of the electron wave, but of the probability of ‘finding’ its associated electron.* The alternating peaks and troughs of the electron wave translate into a pattern of quantum probabilities—in this location (which will become a bright fringe) there’s a higher probability of finding the next electron, and in this other location (which will become a dark fringe) there’s a very low or zero probability of finding the next electron.

  Just think about what’s happening here. Before an electron strikes the screen, it has a probability of being found ‘here’, ‘there’, and ‘most anywhere’ where the square of the wavefunction is bigger than zero.

  Does this mean that an individual electron can be in more than one place at a time? No, not really. It is true to say that it has a probability of being found in more than one place at a time and there is definitely a sense in which we think of the electron wavefunction as delocalized or distributed. But if by ‘individual electron’ we’re referring to an electron as a particle, then there is a sense in which this doesn’t exist as such until the wavefunction interacts with the screen, at which point it appears ‘here’, in only one place, as shown in Figure 5b.

  That this might be a bit of a problem was recognized in the late 1920s/early 1930s by John von Neumann. If ‘measurement’ is just another kind of quantum process or transition, then von Neumann argued that this suggests a need for a ‘measurement operator’, such that

  The measurement outcome is then just the expectation value of the measurement operator.

  Just like the distributed interference pattern shown in Figure 5a, the wavefunction in question may consist of different measurement possibilities, such as the pointer of the gauge above pointing to the left or to the right. Von Neumann realized that there is nothing in the mathematical structure of quantum mechanics that explains how we get from many possible outcomes to just one actual outcome. So, to ensure that the structure is mathematically robust and consistent, he had no choice but to postulate a discontinuous transition or jump which gets us from the possible to the actual. This postulate is generally known today as the ‘collapse of the wavefunction’. It is absolutely central to the ongoing debate about how quantum theory is to be interpreted.

  Quantum probability is not like classical probability

  One more thing. That there’s a 50% probability that a tossed coin will land ‘heads’ simply means that it has two sides and we have no way of knowing (or easily predicting) which way up it will land. This is a classical probability born of ignorance. We can be confident that the coin continues to have two sides—heads and tails—as it spins through the air, but we’re ignorant of the exact details of its motion so we can’t predict with certainty which side will land face up.

  Quantum probability is thought to be very different. When we toss a quantum coin* we might actually be quite knowledgeable about most of the details of its motion, but we can’t assume that ‘heads’ and ‘tails’ exists before the coin has landed, and we look.

  Einstein deplored this seeming element of pure chance in quantum mechanics. He famously declared that ‘God does not play dice’.3

  For a specific physical system or situation, there is no such thing as the ‘right’ wavefunction

  Physics is a so-called ‘hard’ or ‘exact’ science. I take this to mean that its principal theoretical descriptions are based on rigorous mathematics, not on words or phrases that can often be ambiguous and misleading. But mathematics is still a language, and although we might marvel at its incredible fertility and ‘unreasonable effectiveness’,4 if not applied with sufficient care it is still all too capable of ambiguity and misdirection.

  Centuries of very highly successful, mathematically based physics have led us to the belief that this is all about getting the right answer. Nature behaves a certain way. We do this, and that happens. Every time. If the mathematics doesn’t predict that with certainty every time we do this, then we’re inclined to accept that the mathematical description isn’t adequate, and we need a better theory.

  In quantum mechanics, we’re confronted with a few things that might seem counterintuitive. But this is still a mathematically based theory. Sure, we’ve swopped the old classical observables such as momentum and energy for mathematical operators which we use to unlock their quantum equivalents from the box we call the wavefunction. But—to take one example—the frequencies (and hence the energies) of the lines in an atomic spectrum are incredibly precise—just look back at Figure 1. If quantum mechanics is to predict what these should be, then surely this must mean discovering the precise expression for the wavefunction of the electron involved?

  And it is here that we trip over another of quantum mechanics’ dirty little secrets. There is really no such thing as the ‘right’ wavefunction. All we need is a function that is a valid solution of the wave equation. Isn’t this enough to define the ‘right’ one? No, not really. Whilst there are some mathematical rules we need to respect, we can take any number of different solutions and combine them in what’s known as a superposition. The result is also a perfectly acceptable solution of the wave equation.

  I want to illustrate this with an example from the quantum theory of the hydrogen atom, consisting of a nucleus formed by a single proton, ‘orbited’ by a single electron. In fact, this was the problem that Schrödinger addressed in his 1926 paper with such spectacular success. The wavefunctions of lowest energy form spherical patterns around the central nucleus. But there are wavefunctions of modest energy that are shaped like dumbbells. There are three of these.

  All three of these solutions of the wave equation are characterized by a set of quantum numbers. Two of these are the same for each of the dumbbell-shaped functions, but the third differs from one to the other, taking values of –1, 0, and +1, as shown in Figure 6a. For now it doesn’t really matter what these quantum numbers represent. Here’s the thing. Whilst these are the ‘natural’ solutions of the wave equation, they’re not the most helpful when we come to think about combining atoms in three-
dimensional space to form molecules which, after all, is what chemistry is all about.

  Figure 6 The ‘natural’ solutions of the Schrödinger wave equation for the hydrogen atom include a set of wavefunctions characterized by quantum numbers with values +1, 0, and –1. But it’s often more helpful to combine these in the way shown here, which produces three wavefunctions directed along the three spatial dimensions characterized by Cartesian coordinates x, y, and z. So, which are the ‘right’ wavefunctions?

  It’s much easier to deal with wavefunctions defined in three spatial dimensions, using Cartesian x, y, and z coordinates. This is okay for the function characterized by the quantum number 0, as we can simply define this to lie along the z coordinate. But what of the others? Well, this turns out to be relatively easy. To get a wavefunction directed along the y coordinate we form a superposition in which we add the functions corresponding to +1 and –1, as shown in Figure 6b. To get a wavefunction directed along the x coordinate we form a superposition in which we subtract the function corresponding to +1 from the function corresponding to –1. Because we’re combining functions that are known to be solutions of the wave equation, and provided we follow the rules, we can be confident that the superpositions represent valid solutions, too. The resulting functions are shown mapped along the three coordinates in Figure 6c.

  But which then are the ‘right’ wavefunctions? Students learn fairly quickly that there really isn’t a straightforward answer to this question. The ‘right’ wavefunction obviously depends on what kind of system we’re dealing with, but we’re free to choose the form that’s most appropriate for the specific problem we’re trying to solve.

  Delocalized waves can be combined together in ways that localized particles simply can’t, and we can take full advantage of this in quantum mechanics.

  Heisenberg’s uncertainty principle is about what we can know. It is not about what we can only hope to measure

  Werner Heisenberg hated Schrödinger’s wave mechanics. Not because the mathematics was dodgy, but because Schrödinger insisted that the wavefunction be taken literally as the physical description of the electron as a matter wave. The trouble is that waves flow smoothly and continuously, and there is no room in this picture for sudden quantum jumps of the kind needed to interpret the transitions that form an atomic spectrum. Schrödinger simply doubled down and denied that the jumps happened at all, proclaiming: ‘If all this damned quantum jumping were really here to stay, I should be sorry I ever got involved with quantum theory.’5

  In 1927, Heisenberg realized that the essential discontinuity—the ‘jumpiness’—at the heart of quantum mechanics implies a fundamental limit on what we can discover about the values of pairs of physical observables, such as position and linear momentum. At first, he believed that this limit arises because of an inevitable ‘clumsiness’ involved in making measurements on delicate quantum systems with our large-scale, laboratory-sized instruments. For example, Heisenberg argued, if we want to determine the precise position of an electron in space we need to locate it by hitting it with photons of such high energy that we must forgo any hope of determining the electron’s precise momentum. Any attempt to ‘see’ where the electron is (or, at least, was) will just knock it for six, preventing us from seeing to where and how fast it was going.

  But Bohr disagreed, and the two argued, bitterly. Bohr insisted that the uncertainty principle has nothing to do with the clumsiness or otherwise of our measurements. Instead it implies a fundamental limit on what we can know about a quantum system.

  Perhaps the simplest way of explaining Bohr’s point relies on the essential duality of waves and particles in quantum mechanics. Think about how we might measure the wavelength of a wave. We could infer the wavelength by counting the numbers of peaks and troughs in a certain fixed region of space. Each wave cycle consists of one peak and one trough, and the wavelength is the distance from the start to the finish of the cycle. So we sum the numbers of peaks and troughs, and divide by two. This tells us the number of cycles in our spatial region. The wavelength is then the length of this region divided by the number of cycles.

  Obviously, we will struggle to make any kind of precise measurement if our sample region is shorter than the wavelength. We quickly realize that we can increase the precision by making the region large enough to include lots and lots of cycles. Now, from the de Broglie relation, a precise wavelength gives us a precise measure of linear momentum for the associated particle. But, of course, we’ve deliberately made our sample region large, so we’ve lost any hope of measuring a precise position for the associated particle. It could be anywhere in there.

  The opposite is also true. It’s possible to add together a large number of waves in a superposition, each with a different wavelength, such that the wave has a single peak located at a very precise position. Such a superposition is called a ‘wavepacket’. This gives us a fix on the precise position, but now we’ve lost any hope of measuring a precise momentum, because our wavepacket consists of a broad range of wavelengths, implying a broad range of momenta.

  Bohr’s view prevailed, and we now write Heisenberg’s uncertainty relation as*

  Note that nowhere does this say that measurements of position and momentum are somehow mutually exclusive. We can in principle measure the position with absolute precision (zero uncertainty), but then the momentum would be completely undetermined: it would have infinite uncertainty. There’s nothing preventing us from determining both position and momentum with more modest precision within the bounds of the uncertainty principle.

  The principle is not limited to position and momentum. It applies to other pairs of observables. Perhaps the best known relates to energy and time, which we will meet again in Chapter 4. Now, there’s a caveat. There are arguments that the energy–time uncertainty relation actually doesn’t exist except as a variation of that for position and momentum. Early attempts to derive the energy–time relation from first principles proved rather unsatisfactory. To my knowledge, the most widely accepted derivation, published in 1945 by Leonid Mandelstam and Igor Tamm, clearly specifies the interpretation of ‘time’ in the relation as a time interval.

  I think this is enough for now. I want you to be clear that what I’ve described so far is based on the ‘authorized’ or ‘official’ version of quantum mechanics taught to science students all around the world. We’ll see in Part II that as the search for meaning has unfolded in the past 90 years or so, some physicists and philosophers have happily challenged this authority, and we shouldn’t assume that the version taught today will still be taught in another 90 years’ time.

  Eager readers will also note that I’ve deliberately held back some of the more infamous examples of quantum weirdness—such as Schrödinger’s cat and the Einstein–Podolsky–Rosen experiment. Please be patient: we will come to these in Chapter 4. I first want to give you some context in which to think about them.

  To summarize, we’ve seen that experimental discoveries in the first decades of the twentieth century led to the realization that physical reality is inherently lumpy. In the classical mechanics of everyday life, we can safely ignore Planck’s constant and the lumpiness it implies, and assume everything is smooth and continuous. But at the level of molecules, atoms, and subatomic particles, from which everything in the visible universe is constructed, Planck’s constant comes into its own and we can no longer ignore the duality of waves and particles.

  De Broglie opened Pandora’s Box in 1923. Schrödinger gave us his wave equation and his wavefunctions a few years later. The all-too-familiar observables of classical mechanics became locked away inside the quantum wavefunction, requiring mathematical operators to liberate them from their prison. Born said that the wavefunctions are utterly inscrutable; they tell us only about quantum probabilities. Heisenberg (and Bohr) explained that the heart of quantum mechanics beats uncertainly. Nature suffers a peculiar arrhythmia.

  And then the debates began. What is quantum mechanics telling
us about the nature of physical reality? And just what is this thing called reality, anyway?

  * I’ve put ‘orbiting’ in inverted commas because the electron doesn’t orbit the nucleus in the same way that the Earth orbits the Sun. In fact, it does something a lot more interesting, as we’ll soon see.

  * ‘Black-body’ doesn’t refer in any way to the colour of the walls of the cavity, but rather to the way they absorb and emit the radiation trapped inside. In theory, a ‘black’ body absorbs and emits radiation ‘perfectly’, meaning that the radiation doesn’t depend on what the walls are made of.

  * However, look closely and you’ll see that the edges of this band show a distinct diffraction pattern of alternating light and dark ‘fringes’.

  † The cloud chamber was invented by Charles Wilson. It works like this: an energetic, electrically charged particle passes through a chamber filled with vapour. As it passes, it dislodges electrons from atoms in the vapour, leaving charged ions in its wake. Water droplets condense around the ions, revealing the particle trajectory.

  * In classical mechanics, linear momentum derives from the uniform motion of an object travelling in a straight line, calculated as the object’s mass × velocity. However, in quantum mechanics, the calculation of linear momentum is rather different, as we’ll see very soon.

  * The square root of minus 1 is an ‘imaginary number’, usually written as i. This might seem obscure, but it crops up all the time in mathematics and physics. All you need to remember is that i2 = –1.