Wavefunctions, Probability & Uncertainty

Ψ is complex, and no instrument reads a complex number, so Ψ is not the thing you measure. What you measure is |Ψ|² = Ψ*Ψ, which is real, never negative, and — this is the part that trips people — a density rather than a probability. In one dimension Ψ carries units of m^(−1/2) precisely so that |Ψ|² comes out in m⁻¹, a probability per metre. The optical analogy helps as far as it goes: a light wave's field E is not what a screen records, the irradiance goes as E², and for single photons that irradiance is the arrival probability. One more feature comes free. Multiplying the whole of Ψ by e^(iα) leaves |Ψ|² untouched, so an overall phase is unobservable. A phase difference between two parts of Ψ is a different matter entirely, and the rest of this lesson turns on it.

A candidate wavefunction usually arrives with its shape known and its height free. The particle has to be found somewhere, so ∫|Ψ|² dx = 1 taken over all x, and that single condition fixes the height. Take the ground state of a box of width L, Ψ₁ = A sin(πx/L) inside and zero outside. Then ∫₀ᴸ A² sin²(πx/L) dx = A²L/2, using the average value ½ of sin² over a half-cycle, so A²L/2 = 1 and A = √(2/L). For L = 1.00 nm that is A = √(2 ÷ 1.00 × 10⁻⁹ m) = 4.47 × 10⁴ m^(−1/2) — an odd-looking unit that is exactly right, since A² times a length must be dimensionless. Normalisation also disqualifies candidates: a pure plane wave e^(ikx) has |Ψ|² = 1 everywhere and an infinite integral, so it is a useful idealisation of a beam, not a state of one particle.

At the centre of that 1.00 nm box the density is |Ψ₁|² = 2/L = 2.0 × 10⁹ m⁻¹. A number vastly larger than 1 is no contradiction, because it is a probability per metre and only becomes a probability once multiplied by a length. Over a window Δx narrow enough that the density hardly changes, P ≈ |Ψ(x)|² Δx: a 0.020 nm window at the centre holds 2.0 × 10⁹ m⁻¹ × 2.0 × 10⁻¹¹ m = 0.040, a 4.0% chance. For a wider window you must integrate. The middle tenth of the box, from 0.45L to 0.55L, gives ∫|Ψ₁|² dx = 0.198, not the 0.100 of an evenly spread particle, because the ground-state density is humped in the middle. Two habits follow: check that a probability you have quoted lies between 0 and 1, and never compare two densities without saying over what width.

Send electrons one at a time at a double slit. If Ψ₁ and Ψ₂ are the amplitudes at a screen point P for the two openings, the amplitude there is Ψ₁ + Ψ₂ and the arrival rate goes as |Ψ₁ + Ψ₂|² = |Ψ₁|² + |Ψ₂|² + 2|Ψ₁||Ψ₂| cos φ, where φ is the phase difference set by the path difference. Classical bookkeeping — this electron went through slit 1 or slit 2, so add the two probabilities — keeps only the first two terms and predicts a featureless blur. The cross term is the fringes. With equal amplitudes it swings the rate from 4|Ψ₁|² at a maximum to exactly zero at a minimum, while the classical sum sits flat at 2|Ψ₁|²; averaged across the screen the two agree, so interference redistributes electrons without creating them. Arrange any measurement that reveals which slit was used and φ stops being defined, the cross term averages away, and the fringes go with it.

One electron makes one mark. It does not arrive smeared over the screen in the shape of |Ψ|², and nothing in the theory says which mark it will make. The distribution appears only when the same preparation is repeated thousands of times and the marks are histogrammed, which is why an experiment quoting a probability must say what the preparation was. The result also changes the state. Detect the electron inside a 0.020 nm window and an immediate second measurement finds it in that window again with near certainty: the state that follows the detection is one localised there, not the spread-out Ψ you started with. So Δx is not a confession of ignorance about a particle that secretly had a position. It is the width of the distribution the preparation produces, and preparing a narrower one is a physical act with a price.

That price is the uncertainty relation, and its origin is wave analysis rather than clumsiness. A single wavelength extends forever; to build a bump of width Δx you must add waves over a range of wavenumbers Δk, and the mathematics of Fourier sums gives Δx Δk ≥ ½. Each component carries p = ℏk, so Δx Δpₓ ≥ ℏ/2. A Gaussian packet is the one shape that reaches equality; every other shape does worse. Put numbers to it: hold an electron inside Δx = 0.10 nm, about an atomic radius, and Δpₓ ≥ 1.055 × 10⁻³⁴ ÷ (2 × 1.0 × 10⁻¹⁰) = 5.3 × 10⁻²⁵ kg m s⁻¹, a speed spread of 5.8 × 10⁵ m s⁻¹ and a kinetic energy scale near 0.95 eV. Atoms are stable at the size they are because squeezing the electron further costs more kinetic energy than the extra Coulomb attraction returns.