The Born Interpretation and Probability Density

Introduction

A wavefunction can be positive, negative or complex, so it cannot itself be a probability. In 1926 Max Born proposed the rule that connects ψ to experiment: the square modulus of the wavefunction gives the probability of finding the particle at a particular place. This single idea turned Schrödinger's waves from mysterious mathematical objects into testable predictions, and it is the reason that orbital pictures show "where the electron is likely to be".

Core explanation

The rule in one dimension. For a particle moving along x with wavefunction ψ(x), the probability of finding it between x and x + dx is

P = ψ(x) ² dx = ψ (x)ψ(x) dx

The quantity ψ(x) ² is the probability density : probability per unit length. It is not itself a probability; it becomes one only when multiplied by a length.

Finite regions. For a region from x = a to x = b, the probabilities of all the tiny intervals are added, which means integrating:

P(a ≤ x ≤ b) = ∫ₐᵇ ψ(x) ² dx

If the interval δx is small compared with the distance over which ψ changes, the integral is well approximated by ψ(x) ² δx.

Three dimensions. For a particle in space, the probability of finding it in a volume element dτ at position r is ψ(r) ² dτ. In Cartesian coordinates dτ = dx dy dz. For atoms, spherical polar coordinates are more natural, and then dτ = r² sin θ dr dθ dφ. The factor r² is important: a thin shell far from the nucleus contains much more volume than one close to it, which is why the most probable distance of an electron can differ from the point of greatest probability density.

Units. A probability is dimensionless. In one dimension ψ ² must have units of m⁻¹, so ψ has units of m⁻¹ᐟ². In three dimensions ψ ² has units of m⁻³ and ψ has units of m⁻³ᐟ². Checking units is a useful way of catching errors in wavefunction expressions.

What the Born interpretation does not say. It does not claim that the electron is smeared out like a cloud of charge that is partly here and partly there. A single measurement of position gives one definite result. The Born rule predicts the distribution of results from many identical measurements on identically prepared systems. In chemistry, however, it is often convenient to multiply ψ ² by the electron charge and treat the result as an electron density — a useful and legitimate description of the average charge distribution.

Nodes. Where ψ = 0, the probability density is zero: the particle will never be found exactly there. Such points, lines or surfaces are nodes . They arise from the wave nature of the particle and become more numerous in higher-energy states.

Consequence for wavefunctions. Because the particle must be somewhere, the total probability over all space must equal one: ∫ ψ ² dτ = 1. This requirement, normalisation, is the subject of the next page.

Formulae

P(x to x + dx) = ψ(x) ² dx. P(a ≤ x ≤ b) = ∫ₐᵇ ψ ² dx. P(in dτ) = ψ ² dτ, with dτ = dx dy dz = r² sin θ dr dθ dφ. ψ ² = ψ ψ.

Step-by-step reasoning

To find the probability of locating a particle in a small region:

1. Evaluate ψ at the centre of the region. 2. Square its modulus to obtain ψ ². 3. Multiply by the size of the region (length or volume) in consistent units. 4. Check that the answer is dimensionless and lies between 0 and 1. 5. If the region is not small, integrate ψ ² instead.

Visual explanation

Draw ψ for a particle in a box as a single hump of a sine curve. Shade a narrow strip at the centre and an equally narrow strip near a wall. The strip at the centre, where ψ ² is tallest, has the larger area and hence the larger probability.

Real-world analogy

A population-density map shows people per square kilometre. To estimate how many people live in a district you multiply the density by the district's area. ψ ² works in the same way: it is a density that must be multiplied by a length or volume to give a probability.

Real-world example

X-ray crystallography measures electron density in crystals. The maps it produces — contours showing where electrons are concentrated around each atom and along bonds — are experimental pictures of the sum of ψ ² over all occupied orbitals, used routinely to locate atoms in new compounds and proteins.

Why?

Why does the Born interpretation use ψ ² rather than ψ? Probabilities must be real and non-negative, whereas ψ can be negative or complex. Only ψ ψ is guaranteed to be real and non-negative, and it also lets waves interfere before the square is taken.

Common misconception

" ψ ² is the probability of finding the particle at a point." The probability at an exact point is zero; ψ ² is a density that gives a probability only when multiplied by a finite length or volume.

Worked example

Question: The hydrogen 1s wavefunction is ψ = (1/πa₀³)^½ e^(−r/a₀), with a₀ = 52.9 pm. Find the probability of finding the electron in a cube of volume 1.0 pm³ (a) centred on the nucleus and (b) at a distance a₀ from it.

Reasoning: (a) At r = 0, ψ ² = 1/(πa₀³) = 1/(π × 1.48 × 10⁵ pm³) = 2.15 × 10⁻⁶ pm⁻³. P = 2.15 × 10⁻⁶ × 1.0 = 2.2 × 10⁻⁶. (b) At r = a₀ the density is multiplied by e⁻² = 0.135, giving P = 2.9 × 10⁻⁷.

Answer: (a) 2.2 × 10⁻⁶; (b) 2.9 × 10⁻⁷.

Quick check

1. What are the units of a one-dimensional wavefunction ψ(x) when x is in metres? Answer: m⁻¹ᐟ², so that ψ ² dx is a dimensionless probability.

Exam focus

Examiners look for the phrase " ψ ² dτ is the probability of finding the particle in the volume element dτ". Show the multiplication by a length or volume explicitly, and remember the r² factor when working in spherical polar coordinates.

Advanced insight

For a system of several electrons the Born rule is applied to the whole many-electron wavefunction: ψ(r₁, r₂, …) ² dτ₁ dτ₂ … is the joint probability of finding electron 1 in dτ₁, electron 2 in dτ₂ and so on. The one-electron density that chemists plot is obtained by integrating over all but one electron's coordinates.

Summary

Born's interpretation states that ψ ² dτ is the probability of finding a particle in the volume element dτ. ψ ² is a probability density, with units that make the product dimensionless. Probabilities over finite regions come from integration. Nodes are places of zero probability. Because the particle must be somewhere, the total probability over all space is one.

Practice questions

1. For a particle in a box of length L = 1.00 nm with ψ = (2/L)^½ sin(πx/L), find the probability of finding it within 0.01 nm of the centre of the box, i.e. in an interval δx = 0.02 nm. Answer: At x = L/2, ψ ² = 2/L = 2.00 nm⁻¹, so P ≈ 2.00 × 0.02 = 0.04. 2. For the same wavefunction, what is the probability density at x = 0? Explain. Answer: Zero, because sin(0) = 0; the wall is a node where the particle is never found. 3. Write the volume element in spherical polar coordinates and explain why the r² factor matters for atoms. Answer: dτ = r² sin θ dr dθ dφ; the r² factor means shells at larger r contain more volume, so radial probabilities are weighted towards larger distances. 4. Using the 1s wavefunction, by what factor is the probability density at r = 2a₀ smaller than at the nucleus? Answer: ψ ² is proportional to e^(−2r/a₀), so the factor is e⁻⁴ ≈ 0.018.