The Wavefunction and the Born Probability Rule
For a single particle in a pure state, a position wavefunction gives complex amplitudes. Their squared magnitude gives position probability density. Integrating this density over a region predicts the probability of detecting the particle there.
What a learner can do afterwards
- States that the squared magnitude of the wavefunction gives position probability density and explains how interval probability is obtained
- Interprets an interference pattern built one particle at a time
- Distinguishes the wavefunction itself from the probability density it produces
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Quantum mechanics uses a wave function, written psi, for a single particle in a pure state. Here we consider position along a line and leave spin out. Psi is a complex-valued mathematical description used to predict measurements, including where the particle may be detected. It is not something you can see or touch. Quantum mechanics also explains phone chips and lasers.
The Born rule gives position density |psi(x)|², the squared magnitude of psi. Probability in an interval is the area under this density, found by integrating over that interval. For constant density, this area is a rectangle: density times width. Multiplying density by width is an approximation when the density varies little.
A normalized wave function has a total position probability of one. Integrate |psi(x)|² over all space to find that total. Normalization expresses certainty that a detection position is somewhere in space, not a specific position in advance.
Send identically prepared electrons through two open slits without which-slit information. Keep only one electron in the apparatus at a time. Each detection gives one dot. Psi does not predict its exact position. Many dots build an interference pattern described by the squared magnitude of the combined amplitude. The pattern does not require several electrons to be present together.
With both slits open and no which-slit information, add the amplitudes from the two slits before taking the squared magnitude. Amplitudes can add or cancel. Adding the separate single-slit probabilities misses this interference. The Born rule predicts detection probabilities; it alone does not settle whether a particle had a definite unmeasured path.
Compare interval probabilities using their areas, not just density at one point. For constant density, probability equals density times width. For two equal-width intervals, the one with higher constant density has greater probability and is expected to collect more detections over many identical preparations. Finite totals can fluctuate.
Take the squared magnitude of psi for position density. Integrate over an interval to find its probability; normalized density integrates to one over all space.
Lesson corrections and replacement questions authored for this dataset.
The rewritten readings were written for this dataset. Study cards corrected for this dataset. The linked TED-Ed video is credited to Chad Orzel.
More to read
From a wave function to position probabilities
A wave function, psi, represents a single particle's pure state mathematically and predicts measurement results. In one dimension its variables are position and time; its values can be complex.
Under the Born rule, position probability density equals |psi|². Use the squared magnitude, which need not equal psi². Integrating the density over an interval gives that interval's probability, or the area beneath the density curve. Multiplying density by width works exactly for constant density and approximately for small variations. Normalization makes the integral across all space equal to one.
Individual detections and quantum patterns
Each position measurement detects an identically prepared particle at one location. Repeating this many times builds a distribution governed by |psi|². The curve gives density at a position; an interval's probability requires its area.
With two coherent slits and no information about which slit an electron takes, combined amplitudes yield interference. Interference can build up even if each electron leaves the apparatus before the next enters. An atomic orbital likewise specifies a position probability distribution, not a classical path. Such predictions do not settle how to interpret a particle's behaviour between measurements.
Evidence for waves of matter
De Broglie's proposal assigns matter a wavelength: Planck's constant divided by momentum. Crystal experiments supported it when electron scattering showed diffraction patterns matching calculations with that wavelength.
Quantum wave functions mathematically predict measurement results. Diffraction links familiar wave behaviour with the probability patterns calculated using quantum amplitudes.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.