The Wavefunction & Born's Rule

Chapter 7 left us a rule and chapter 8 a bargain, both written with a symbol we never properly defined. Electrons arrived as single dots that built interference stripes; squeezed wave packets bulged in momentum. The thing that interferes and bulges is the wavefunction, ψ. For one particle moving along a line, ψ(x) attaches a number to every position x, and the number is complex: it has a real part and an imaginary part or, in the picture of chapter 2, it is an arrow with a length and a direction. The arrow's length is called the magnitude and its direction the phase.

Physicists call ψ(x) the probability amplitude for finding the particle at x. The word amplitude is chosen with care: ψ is not itself a probability but the raw material from which probabilities are made. Complex numbers are needed because the phase carries physical information. A particle with a perfectly definite momentum p is described by arrows of identical length everywhere whose direction turns steadily along x, completing one full turn every de Broglie wavelength λ = h/p. Its real part undulates like a cosine and its imaginary part like a sine, yet its magnitude is the same at every point. The winding phase is the momentum; the constant magnitude says the particle is equally likely to be found anywhere.

Chapter 6 insisted that nothing material ripples in a matter wave. When Erwin Schrödinger published his wave equation in 1926 he hoped otherwise, proposing that the electron really is a continuous cloud of charge with density proportional to |ψ|², spread through space like jam. Max Born recalled in his Nobel lecture that to him and his colleagues in Göttingen this seemed unacceptable, because experimenters could already count particles one at a time with scintillation screens and Geiger counters, and photograph their individual tracks in cloud chambers. Detectors never register three-tenths of an electron. Notice, too, that ψ itself is not a density of anything: for one particle on a line it is measured in metres to the power minus one-half, and only its square has the units of a probability per metre.

Born's rule, in its modern form for position, says this. Square the magnitude of the amplitude and you have the probability density, |ψ(x)|²: the probability per unit length of finding the particle near x when its position is measured. To square a complex number's magnitude, add the squares of its real and imaginary parts or, equivalently, multiply it by its complex conjugate, the mirror-image arrow with its angle reversed. Either way the result is a length squared, so a wavefunction that is negative or imaginary somewhere can never produce a negative probability: sign and phase vanish in the squaring.

A density is not yet a probability, any more than a population density of 2,000 people per square kilometre tells you how many people live on your street. You need a size of region. For a narrow window of width Δx around x, the probability of finding the particle there is approximately |ψ(x)|² times Δx. For a wider region, from a to b, slice it into thin strips, find the probability of each, and add them up. In the limit of infinitely thin strips that sum becomes the area under the curve of |ψ|² between a and b, which mathematicians call the integral. Probability, in Born's rule, is area.

Try it on the simplest bound state. An electron trapped between hard walls one nanometre apart, in its lowest-energy state (chapter 11 derives it), has ψ = √(2/L)·sin(πx/L), so its density peaks at the centre at 2 per nanometre. The chance of finding it within 0.05 nm of the centre is therefore about 2 per nanometre × 0.1 nm = 0.20, and the exact area is 0.198. The same squaring recipe governs energy, spin and every other measurable quantity (chapters 14, 15 and 17). One caution: the rule is a postulate of quantum mechanics, not a consequence of Schrödinger's equation, and attempts to derive it from deeper principles, notably within the many-worlds interpretation, remain debated.

Measure the particle's position across the whole line and you will find it somewhere, every single time. The probabilities of all the places must therefore add up to exactly 1: the total area under |ψ|² must equal 1, and a wavefunction that meets this condition is said to be normalized. Solving a wave equation usually gives a shape without fixing its overall size, because any solution multiplied by a constant is still a solution. Normalization fixes the size in two steps: find the total area under the square of your unnormalized shape, then divide the shape by the square root of that area. The square root appears because probabilities, not amplitudes, must total 1.

Here is the recipe on a smooth hump that vanishes at two walls a distance L apart: ψ(x) = A·x(L − x) between the walls and zero outside, with A to be found. The area under x²(L − x)² from 0 to L works out to L⁵/30, so normalization demands A² times L⁵/30 = 1, giving A = √(30/L⁵). For L = 1 nanometre, A is about 5.48 in units of nanometres to the power minus five-halves. The density then peaks at the centre at 1.875 per nanometre, the middle half of the gap holds 79.3 percent of the probability, and each outer quarter 10.35 percent.

Not every wave can be normalized. The endless wave of definite momentum from Part 1 has a density that is the same everywhere, so the area under it is infinite. Physicists use such plane waves as idealizations, limits of very long wave packets, but a real particle is always a normalizable packet. Once normalized, a wavefunction stays normalized: Schrödinger's equation, next chapter's subject, carries ψ forward in time without creating or destroying total probability. Normalization does leave one freedom untouched: multiplying ψ by −1, or by any complex number of size 1, changes no area at all, a loose end Part 5 ties up.

Born's rule predicts probabilities: nothing less, and nothing more. Prepare a thousand electrons in exactly the same state and measure each one's position. Every measurement returns a single, definite place, and quantum mechanics cannot say in advance which place it will be. Yet the fraction found in any region converges, as runs pile up, onto the area under |ψ|² for that region, just as Akira Tonomura's 1989 electron experiment showed in chapter 7. In standard quantum mechanics the randomness is fundamental, not a record of details we failed to track, although some interpretations dispute that. Einstein never accepted it: on 4 December 1926 he wrote to Born that he, at any rate, was convinced the Old One does not play dice.

The sharpest prediction for an ensemble, a large collection of identically prepared systems, is its average. The expectation value of position, ⟨x⟩, is the average position weighted by the probability density: multiply each position by |ψ|² there and take the total area. It is the long-run mean of many measurements, not a forecast for any single one, and it need not be a place the particle can be found. The second state of the one-nanometre box has a node, a point where |ψ|² vanishes, exactly at the centre, which is also where ⟨x⟩ sits: only 0.65 percent of detections land within 0.05 nm of the average position. The spread is the standard deviation σ: the square root of the mean of the squared positions minus the square of the mean, the very Δx of chapter 8.

How exact are the statistics? The average of N measured positions wanders about ⟨x⟩ by roughly σ/√N, so for a packet with σ = 0.09 nm ten thousand detections pin the mean down to within about 0.0009 nm. Born's rule itself has been put on trial. In 1994 Rafael Sorkin noted that if probabilities come from squaring sums of amplitudes, interference only ever involves pairs of paths, so a three-slit pattern is fixed entirely by the single-slit and two-slit patterns. In 2010 Urbasi Sinha, Gregor Weihs and colleagues sent photons through three slits and bounded any genuinely three-path term below one percent of ordinary two-path interference, and in 2017 Weihs's group in Innsbruck pushed the bound below one part in ten thousand with a five-path interferometer. Born's footnote has held.

Multiply a wavefunction everywhere by the same complex number of size 1, such as −1 or i, and every arrow turns through the same angle while keeping its length. This is a change of global phase, and since |ψ|² depends only on lengths, every density, every area and every expectation value stays exactly as it was. No experiment can tell ψ from −ψ; they are two labels for one physical state. The Born Rule Detector's phase-flip button does exactly this, and changes nothing you can measure. Squaring hides the phase of a single amplitude completely.

Relative phase is another matter. When a wavefunction is a sum of two parts, ψ₁ + ψ₂, the density, |ψ₁ + ψ₂|², contains chapter 7's interference term, which depends on the angle between the two arrows at each point. Rotate one part relative to the other and probability moves. Put an electron in the one-nanometre box in the state 0.6ψ₁ + 0.8ψ₂, where ψ₁ and ψ₂ are the two lowest box states and 0.6² + 0.8² = 1 keeps it normalized. The second state is positive in the left half of the box and negative in the right, so it reinforces the first on the left and partly cancels it on the right: 90.7 percent of detections land in the left half. Flip the sign of ψ₂ alone and only 9.3 percent do. Same ingredients, same proportions, opposite outcome.

Phase can also vary from place to place within a single wave, and then it encodes motion. Take a packet whose arrows all point the same way and twist them so that the phase winds one full turn every distance λ along x. At that instant |ψ|² is untouched, but the packet now carries an average momentum of h/λ and sets off moving, while its untwisted twin stays centred and merely spreads. That is why the state of a particle is the full complex ψ, not merely |ψ|²: the density says where the particle may be found now, and the phase says where the density is going.

For a single particle the jam picture is tempting, because ψ(x) can be drawn as a curve over ordinary space. Add a second particle and the picture breaks. Two particles on a line are described not by two waves on that line but by one wavefunction Ψ(x₁, x₂), which assigns an amplitude to every pair of positions: particle one at x₁ and particle two at x₂. It is a function on a plane, not on a line, and |Ψ(x₁, x₂)|² is a joint probability density, the probability per unit area of finding the first particle near x₁ and the second near x₂. In three dimensions two particles need six coordinates and N particles need 3N. The space of all possible arrangements is called configuration space, and that, not the room you are sitting in, is where the wavefunction lives.

Three consequences follow. First, Ψ has no value at a point of ordinary space, only at a whole arrangement, so it cannot be a substance spread through the room. Second, Ψ need not split into one wave per particle; when it does not, the particles are entangled, and what is found for one is correlated with what is found for the other (chapter 20). Third, the bookkeeping explodes. Store Ψ on a grid of 100 points along each axis and one electron needs a million complex numbers, two electrons a million million, and fourteen electrons 10⁸⁴, more than the roughly 10⁸⁰ atoms in the observable universe. That is why quantum chemistry lives on clever approximations, and why Richard Feynman proposed in 1981 to simulate quantum systems with quantum machines.

Experiments read configuration space one sample at a time. Quantum gas microscopes, pioneered by Markus Greiner's group at Harvard in 2009, photograph individual atoms held in a lattice of laser light; each picture is a single draw of where the whole crowd was found together, one point in a space of enormous dimension, and only many pictures reveal the joint probabilities. What Ψ is, physically, remains open. Many-worlds and pilot-wave theorists treat it as real; others regard it as a tool for computing odds. A 2012 theorem by Matthew Pusey, Jonathan Barrett and Terry Rudolph shows, under stated assumptions, that it cannot be a mere summary of ignorance about some deeper state. Whatever it is, Born's rule is how we use it.