Probability amplitudes and superposition

Classical probability has a rule for alternatives: if an outcome can be reached two ways, add the two probabilities. Quantum mechanics keeps that arithmetic and changes the objects it acts on — complex amplitudes in place of non-negative probabilities — with the squaring moved to the end. The consequence is not a small correction. Take a balanced two-path interferometer whose arms deliver amplitudes of equal modulus to detector D₁, and tune them to arrive antiphase: they cancel and D₁ never fires. Now block one arm. D₁ starts firing, at a quarter of the incident rate. Removing a way for the particle to arrive has made arrivals more likely, and no assignment of non-negative probabilities to the two paths can reproduce that, because a sum of non-negative terms cannot fall when a term is added to it. That is the sharp test, and probability-adding fails it while amplitude-adding passes. Note what is not being claimed. The amplitudes belong to the alternatives, not to a particle that somehow traversed both arms, and the cancellation happens in the arithmetic of the state, not in a collision between two halves of an electron.

Multiply out the square and the extra term appears on its own. Writing each amplitude in polar form, ψⱼ = √Pⱼ e^(iθⱼ), we get |ψ₁ + ψ₂|² = (ψ₁ + ψ₂)* (ψ₁ + ψ₂) = |ψ₁|² + |ψ₂|² + ψ₁*ψ₂ + ψ₂*ψ₁ = P₁ + P₂ + 2Re(ψ₁*ψ₂) = P₁ + P₂ + 2√(P₁P₂) cos φ, with φ = θ₂ − θ₁. Three consequences follow. The factor of 2 is not decoration: it counts both cross products in the expansion, and dropping it is the commonest slip in this topic. The term is bounded by ±2√(P₁P₂), so it is largest for balanced paths and dies when either path is blocked. And it depends only on the difference of the phases, so multiplying the whole state by e^(iα) changes nothing physical. With P₁ = 0.09 and P₂ = 0.01 the cross term ranges over ±0.06, giving a maximum of 0.16, a minimum of 0.04 and a visibility of 0.12/0.20 = 0.60. Integrated over the whole screen the cross term averages to zero: it moves probability from the dark fringes into the bright ones, and never manufactures any.

The phase is not a free parameter to be fitted; it is accumulated along the path. For a free particle the action over a path length ℓ is pℓ, so φ = ΔS/ħ = pΔℓ/ħ = 2πΔℓ/λ_dB, with λ_dB = h/p. Every geometrical feature of an interference experiment enters at this point. In a two-slit geometry with slit separation d, the path difference to a screen point at angle θ is d sin θ, so maxima sit where d sin θ = mλ. Put numbers on it: 50 keV electrons have λ_dB = 5.4 pm, so slits 100 nm apart throw fringes λL/d = 27 µm apart on a screen 0.50 m away — a pattern you can record, generated by path differences of a few picometres. Phase also accumulates in time, as e^(−iEt/ħ), so an arm held at a different potential energy for a while arrives shifted; that is the scalar Aharonov–Bohm effect, and it is what neutron interferometers exploit to read a gravitational phase across a height difference of a few centimetres. The magnetic Aharonov–Bohm phase is stranger, and worth keeping straight: it is qΦ/ħ from the enclosed flux Φ, picked up along a route on which the field and the force are both zero, so no potential energy is involved at all.

The rule carries a condition, and the condition is about the rest of the world. Add amplitudes when the two alternatives leave everything else — apparatus, environment, any stray photon — in the same final state. Add probabilities when they leave it in perfectly distinguishable states. Nothing in the condition mentions a person: it is the existence of the record that kills the cross term, not anyone reading it, and a record already buried in a detector's thermal noise counts in full. This is testable, not philosophical. Fullerene molecules interfering in a Talbot–Lau interferometer lose their fringes as the molecules are heated, because a hot C₇₀ emits thermal photons whose wavelength eventually becomes short enough to localise which path it came from; visibility falls as the internal temperature rises, with no mechanical disturbance applied at all. The same condition run in reverse explains identical-particle statistics: exchange paths for two identical particles are indistinguishable in principle, so it is their amplitudes that combine and not their probabilities — with a plus sign for bosons and a minus sign for fermions, and that minus sign is the Pauli exclusion principle.

Between 'no record' and 'perfect record' lies a continuum, and it is quantitative. Let the path tag end in states |d₁⟩ and |d₂⟩. For balanced paths the cross term is multiplied by the overlap, so the visibility is V = |⟨d₁|d₂⟩|, while the best distinguishability the tag can support is D = √(1 − |⟨d₁|d₂⟩|²), and the two obey D² + V² ≤ 1, with equality when both states are pure. Overlap 1 gives V = 1 and D = 0: full fringes, no path information. Overlap 0.6 gives V = 0.6 and D = 0.8, so you can name the path correctly ½(1 + D) = 90% of the time and still watch fringes at 60% depth. Overlap 0 gives V = 0 and D = 1. Notice the second, quieter route to poor fringes: unequal path amplitudes. With P₁ ≠ P₂ the predictability |P₁ − P₂|/(P₁ + P₂) is already non-zero, and since it is the best path guess available it plays the role of D in the same bound, with no marker anywhere in the apparatus. Dim fringes are evidence of path knowledge available in principle, not evidence that a measurement was performed.

This is a calculational postulate with a hole in it. It tells you to add amplitudes when the alternatives are indistinguishable and probabilities when they are not, and it does not say why the existence of a record anywhere should change the arithmetic, nor when a record counts as made. Decoherence fills part of the gap — it shows how fast an environment records path information, and so why fringes are invisible for anything much larger than a molecule — but the step from a spread-out state to one definite click is still postulated separately, and topic 5.6 states it as exactly that. Two further cautions. The rule does not license the sentence 'the particle went through both slits': no amplitude is attached to that claim and no measurement tests it. And the amplitude belongs to a single particle's state, not to a beam. Electrons fired one per second at a double slit, hours apart, still build the complete fringe pattern, so the cross term is never two particles interfering with each other.