物理 · Physics

Wavefunction & Superposition

Day 19 · 2026 · Phase E Quantum Mechanics

Superposition is not "both here and there" — it is a set of arrows that can cancel each other out; ask a different question and the same "superposition" becomes a definite answer.

Fire electrons one at a time, the beam so faint that on average only one is inside the apparatus. Each lands on the screen as one dot — never half a dot. The first few dozen look like scatter. After tens of thousands, a pattern of bright and dark bands grows out of the noise on its own. When Feynman described this apparatus in his 1965 lectures he had to note that nobody had ever actually done it that way; in 1989 Tonomura filmed the whole build-up, electron by electron. In 2002 the readers of Physics World voted it the most beautiful experiment in physics.

One Electron at a Time

Double slit · 1961 / 1989 / 2013

Intuition With one slit open, the screen shows a single blob, brightest in the middle and fading outwards. With both open, common sense says you get two blobs added together, a little brighter in the middle. What you actually get is a row of alternating bright and dark bands — and at some positions that were bright with one slit open, opening the second slit means nothing arrives at all. Adding a route makes fewer things arrive: classical probability cannot write that down under any circumstances.

Mechanism The spacing of the bands is fully calculable. With slit separation d and a distance L from slits to screen, neighbouring bright bands are

Δy = λLd, where λ = hp

Δy reads "delta y" — the gap between two neighbouring bright bands; λ is the Greek letter lambda (not the digit 1, not the letter l) and stands for wavelength; p is momentum and h is Planck's constant. The relation λ = h/p is de Broglie's, from "The Quantum Revolution".

The way the statistics accumulate is the whole point: each electron leaves one whole dot, and the pattern is the distribution of tens of thousands of them. Only one electron is in the apparatus at any moment, so there is no partner to interfere with — it can only interfere with itself.

Three moments in the same run: no single landing is predictable, yet the distribution can be calculated in advance. The blue curve is the measured shape, the red dashed curve is what "add the two probabilities" would give, and the gap between them is interference.

The counterintuitive part The pattern is not collective behaviour among electrons, and the electron is not smeared into a mist: the screen always registers one complete dot. What appears here is a new layer — the single outcome is unpredictable while the distribution is strictly determined, and not because our instruments are too crude.

Cross-disciplinary reading · Engineering / Structural biology

Engineering: electron interference is now a measuring instrument — electron holography turns a magnetic field inside a material into a bending of the fringes, and Tonomura's group used it to image individual flux vortices inside a superconductor.
Structural biology: run it backwards and the position of the fringes reveals the structure. The X-ray diffraction image of 1952 (Photo 51, by Franklin and Gosling) settled that DNA is a double helix because only a helix produces that arrangement of spots.

In one line: the fringes are not built by electrons hitting each other — each electron builds them with itself.

Think: Swap the electron for a whole molecule, hundreds of atoms across. Do the fringes survive?

They do. A football-shaped molecule of 60 carbon atoms (1999) and molecules beyond 25 000 daltons (2019) have both produced interference, given high vacuum and low temperature: nothing along the way may pick up "which slit it went through". The ceiling is not set by size but by coherence.

Probability Amplitudes: Add the Arrows, Then Square

The Born rule · 1926

Intuition Classical probability keeps simple books: two mutually exclusive routes, add the two probabilities, and the total only ever grows. Quantum mechanics inserts a layer underneath. Each route first gets an arrow — a length and a direction. You add the arrows head to tail, and only then does the square of the resulting length become the probability. Two arrows pointing the same way give 4 units of probability; pointing opposite ways, none — while adding the probabilities classically always gives 2.

Mechanism The arrow is called a probability amplitude, and mathematically it is a complex number ψ.

P = |ψ₁ + ψ₂|² = |ψ₁|² + |ψ₂|² + 2|ψ₁||ψ₂| cos Δφ

ψ is the Greek letter psi (say "sigh"); the vertical bars |…| take the length (the modulus) of the arrow; Δφ reads "delta phi" and is the angle between the two arrows, the phase difference. The first two terms are exactly the classical "add the probabilities"; the third is the interference term and is the entirety of what is new — it can be positive or negative, and at its most negative it drives the total down to the digit 0 (zero).

When Born proposed the rule in 1926, the body of the paper said the probability was proportional to ψ; "the modulus squared" arrived as a footnote added at proof stage — and that footnote won him the Nobel Prize in 1954.

The same two routes, one unit of probability each, can combine into 4 units, 2 units or none — it depends entirely on the angle. Classical probability has no such degree of freedom.

The counterintuitive part Only relative phase is ever measurable: rotate every arrow by 30° and no experiment can tell the difference, so the overall phase is pure bookkeeping convention. The angle between arrows, however, is physical — it decides which parts of the screen are bright and which are black. So ψ is neither directly observable nor a disposable intermediate quantity.

Cross-disciplinary reading · Engineering / Computation

Engineering: phased-array radar and noise-cancelling headphones are the classical version of "add amplitudes, then square" — set the phase of each element and the beam points wherever you like with no moving antenna. Classical waves have always added this way; what is new in quantum mechanics is that what gets added are amplitudes of probability.
Computation: amplitudes that cancel are the entire capital of quantum algorithms — Grover's search makes the arrows of the wrong answers cancel while the right one stays aligned. The reverse holds too: simulating quantum systems on classical machines is hard precisely because sampling with weights that can be positive or negative cancels wholesale, which is the still-unsolved sign problem.

In one line: the classical world adds probabilities, the quantum world adds arrows and then squares.

Think: What about three slits — does a genuinely "three-way" kind of interference appear?

Under the Born rule three slits mean three arrows added, and expanding the square leaves only pairwise interference terms; no third-order term can appear. That is a falsifiable prediction, and in 2010 Sinha and colleagues tested it with a three-slit apparatus. The third-order term came out zero to within experimental precision.

Where the Wavefunction Lives

The Schrödinger equation · 1926

Intuition Hand an arrow to every possible position and you have a function ψ(x): a little arrow standing at each point of space, its length carrying probability and its direction carrying phase. How it changes in time is settled completely by the Schrödinger equation — continuous, deterministic, reversible, with no randomness in it anywhere. Randomness only shows up at the step where you take a reading.

Mechanism

i ħ ∂ψ∂t = Ĥ ψ

i is the imaginary unit (i² = −1), and it is precisely what makes ψ a rotating arrow rather than an ordinary real number — only something that can turn has a phase that can cancel; ħ is said "h-bar" and equals h/2π; ∂ψ/∂t is the rate of change of ψ in time (∂ is the partial-derivative sign: let time vary, hold everything else fixed); Ĥ is said "H hat" and is the Hamiltonian, which carries the total energy of the system. The whole line says one thing: energy sets how fast the arrow turns.

One more constraint: the values of |ψ|² over all positions must sum to 1 — the particle has to be somewhere — and the Schrödinger equation happens to keep that sum fixed forever.

The counterintuitive part ψ does not live in the three-dimensional space you and I occupy. The wavefunction of two electrons is ψ(x₁, x₂) — a function of six coordinates; N particles need 3N dimensions. It is not a mist spread through the room but a field of arrows over that enormous space of all possible configurations. Picture it as a cloud in the room and you just about survive the double slit, then fail completely the moment there are two particles.

Cross-disciplinary reading · Chemistry / Computation

Chemistry: molecular orbitals are superpositions of atomic orbitals, and the strength of a bond comes directly from amplitudes reinforcing or cancelling. The "resonance" of a benzene ring is likewise not the molecule flickering between two structures but a single state that is the superposition of both.
Computation: those 3N dimensions are exactly why simulating quantum systems is hard — 50 spins need 2⁵⁰ complex numbers to describe, more than a classical machine can even store. In 1982 Feynman drew the conclusion: then use a quantum system to simulate a quantum system. That is the original motivation for the quantum computer.

In one line: there is no randomness anywhere in the Schrödinger equation — randomness enters at the reading.

Think: Why the square of the modulus, rather than the length itself, or the fourth power?

Because the sum of squares is exactly the quantity Schrödinger evolution preserves: only the sum of |ψ|² stays equal to 1, so only then does probability neither leak away nor appear from nowhere. Gleason went further in 1957: once you demand that probability assignments be consistent across every choice of question, the modulus squared is the only possible rule.

Superposition Is Not "Both at Once"

Choice of basis · the cat, 1935

Intuition Take a spin and ask it "up or down". The answer is up or down. The superposed state is usually described as "it is pointing up and down at the same time". But put the same state to the question "left or right" and the answer is a hundred per cent right. Change the axes and the superposition is gone: superposition is not a property of the state alone, but of the state relative to the set of questions you chose.

The amber arrow is the same state in both panels: on the left it straddles two axes and is called a superposition, on the right it lies along one axis and is definite. The two perpendicular axes stand for two mutually exclusive answers, not for directions in real space.

Mechanism What matters more is telling superposition apart from a mixture. Two boxes: in box A every particle is in the superposed state; in box B each particle is either up or down, half and half, and you simply do not know which. Ask only "up or down" and the two boxes give identical data; ask "left or right" and they part company: A answers right every time, B still gives fifty-fifty. The difference is a measurable experimental fact, not a matter of philosophical taste, and it is called coherence — whether that angle is still there. Let the environment interfere and A degrades into B; that is decoherence, and no "consciousness" takes part in it.

The counterintuitive part Schrödinger's cat of 1935 was sarcasm: his point was that carrying this way of speaking all the way up to a cat becomes absurd. Today "Schrödinger's such-and-such" is fine as a joke about things being undecided and wrong as physics — a real cat collides with air molecules something like 10²⁷ times a second, and coherence does not survive for any measurable length of time.
A more valuable piece of misinformation to dismantle: a quantum computer does not "try all the answers in parallel". n qubits really can hold 2ⁿ branches at once, but you get one measurement and one random outcome; the work is done by interference the algorithm arranged in advance — wrong branches cancelling, right ones lining up. Cancellation does the work, not parallelism.

Cross-disciplinary reading · Language / Engineering

Language: in everyday speech "superposition" is borrowed to mean vague and undecided. The physics is the opposite — it is a completely definite state, one that predicts the outcome of some measurement with certainty: what is vague is not the state but the direction of the question.
Engineering: every machine that lives on phase is rated by its coherence time — superconducting qubits are now in the hundreds of microseconds. Protecting quantum behaviour is not mysticism, it is one instruction: do not let the environment read your phase.

In one line: superposition does not mean "both" — it means you have not yet asked the right question.

Think: If changing axes can make a superposition disappear, could we always pick axes in which nothing is ever superposed?

For a single system, yes — any state is a basis vector of some basis. But once two systems are entangled, no choice of axes leaves both subsystems definite. "Entanglement & Nonlocality" takes apart exactly how that escape route is closed off.

Going Deeper

What does "the electron interferes with itself" actually mean?

Not that the electron splits in half and sends one half through each slit — it is never divided, and the screen always shows one whole dot. The correct reading is that statements about the path have no corresponding fact: as long as the apparatus leaves, in principle, no record of which slit was taken, the amplitudes of the two routes must be added. Feynman pushed this to its limit: every possible path from A to B gets an arrow, and you add them all. That is the path integral. The single classical trajectory is the one whose neighbouring paths have almost the same phase and therefore reinforce, which is precisely where the action is stationary ("The Principle of Least Action"). The classical path is not an exception to quantum mechanics; it is the result of the quantum sum.

The fringes vanish when you detect the path — is that just "looking at it knocks it off course"?

That is the most popular explanation and the most misleading one. Heisenberg's original microscope argument did rely on momentum kicks, but later path markers — tagging an internal energy level of an atom, say — recoil far too little to wash out the fringes, and the fringes disappear anyway. What kills interference is entanglement: once a record exists anywhere in the environment or the apparatus that distinguishes the two routes, they no longer lead to the same final state and the interference term goes to zero, regardless of how gentle the disturbance was. Erase that record and the fringes can be recovered (the quantum eraser, proposed in 1982, realised in 2000) — but only once the data are sorted into coincidence pairs afterwards. Nothing travels backwards in time, and nothing can be signalled faster than light.

Why does the amplitude have to be a complex number — would real numbers not do?

Real numbers give you a little interference (+1 and −1 add to zero) but cannot carry the theory. For states to evolve continuously and reversibly while keeping the total probability at 1, the arrow has to turn continuously in a plane — real numbers offer only two directions, plus and minus, and cannot rotate. The deeper trouble appears when two systems are combined: a quantum mechanics rebuilt over real vector spaces makes different predictions for multipartite experiments. Renou and colleagues turned that into a decidable experiment in 2021, and the results came down on the side of complex numbers. The i is not a convenience of notation; it is load-bearing.

Is the wavefunction a real thing, or only our knowledge of the system?

One camp (ψ-ontic: many-worlds, Bohmian mechanics, spontaneous-collapse models) holds that ψ is a physical entity. The other (ψ-epistemic, such as quantum Bayesianism) holds it is a ledger of belief, so that "collapse" is just an update of knowledge, no stranger than probabilities jumping when you turn a card over. The PBR theorem of 2012 moved the argument substantially: assume that independently prepared systems each carry their own real state, and the purely epistemic reading conflicts with quantum predictions. It did not end the debate, but it removed a large region of the map. The full terrain belongs to "Measurement & Interpretations".

One Electron at a Time

Probability Amplitudes: Add the Arrows, Then Square

Where the Wavefunction Lives

Superposition Is Not "Both at Once"

Going Deeper

Further reading