One electron reaches a detector and leaves a localized event. Another arrives later. After many repetitions, the accumulated events can reveal an interference pattern. The individual detection and the distribution of detections are different features of the experiment, and a successful account must explain both.
The phrase “one particle interferes with itself” names the puzzle memorably. To investigate it, we need to say what was prepared, which alternatives the apparatus made available, and how the probabilities of the final records were calculated.
Events and distributions
A single mark does not display a whole probability distribution. Repeating a controlled preparation allows frequencies to be compared with a predicted distribution. The pattern concerns the relative occurrence of many possible records across those repetitions.
Bach and colleagues' double-slit electron experiment combined control over the slit openings with the recording of individual detection events. Their published study shows the build-up of a diffraction pattern from those events. The connection between discrete detections and an extended statistical pattern is therefore something an apparatus can investigate.
That observation does not determine a unique picture of what the electron “really did” between preparation and detection. It supplies a constraint on such pictures: they must reproduce the observed changes when the arrangement is changed.
A small, complete example
This interactive companion extends the two-output calculation in SMONAD's existing interference guide. Use an ideal balanced two-path interferometer with detectors D0 and D1. It is a simplified two-output model, rather than a simulation of the continuous screen in a double-slit experiment.
Choose a phase convention in which the amplitudes at the two outputs are (1 + eiφ)/2 and (1 − eiφ)/2. Squaring their magnitudes gives
P(D0) = (1 + cos φ)/2; P(D1) = (1 − cos φ)/2.
At zero relative phase, the ideal prediction is D0 with certainty. At a phase difference of π, it is D1 with certainty. Halfway between, at π/2, both outcomes have probability one half. The probabilities always add to one.
Change the phase
Two paths, two possible records. Vary the relative phase. Then allow the paths to leave perfectly distinguishable records in another system.
Phase 0°; no path record: D0 100.0%, D1 0.0%.
These bars show exact probabilities in the stated ideal model, not generated experimental data. With perfectly distinguishable path records left unread, both output probabilities are 50% for every phase.
What a path record changes
Suppose the two paths become correlated with states |e0⟩ and |e1⟩ of another system. The interference contribution now depends on their overlap. Identical record states preserve the interference in this ideal construction. Orthogonal record states make the alternatives perfectly distinguishable and remove the interference from the unconditioned output counts.
The loss does not wait for a person to read the record. It follows from the correlation with another physical system and from considering the detector statistics without conditioning on that system. The checkbox represents that change in the model, not a change in a person's awareness.
If someone later sorts detector events using additional measurements and records, they are examining a different, conditional distribution. Recovering structure in such selected data does not change the earlier unconditioned totals or send a message to the past.
Why probabilities do not simply add
For alternatives that remain coherent, adding amplitudes before squaring produces a cross term. For our outputs, that term transfers probability between D0 and D1 as the phase changes. Replacing the amplitudes by separate path probabilities discards the term and loses the phase dependence.
This is a calculational distinction, not an appeal to a wave metaphor. The Feynman lectures on quantum behaviour introduce the amplitude rules through concrete experimental comparisons. The earlier Reality essay on quantum states gives another example of the information carried by relative phase.
What should a finite count look like?
At a setting where P(D0) is one half, one hundred independent, identically prepared trials have an expected D0 count of fifty. They need not produce exactly fifty. Under the binomial model, the standard deviation is √[Np(1−p)], which is five for N = 100 and p = 1/2.
At ten thousand trials, the expected count is five thousand and the standard deviation is fifty. The absolute fluctuation scale grows, while its size relative to the total falls from 5% to 0.5%. These are model calculations, not guarantees about a particular run. Correlated trials, drifting preparation or detector losses require additional analysis.
The dual-time question
SMONAD's article on temporal entanglement and single-particle interference approaches the phenomenon through the programme's account of temporal generation. The example here supplies a precise target for reading that interpretation.
Ask how the proposed account represents relative phase, how it distinguishes coherent alternatives from retained path records, and how it obtains the two output probabilities. The published phase-selection paper provides a mathematical entry into the question of compatible complex structure. A fuller physical account must connect that structure to the actual preparation and readout operations.
Sources and further reading
- Bach, Pope, Liou and Batelaan (2013), “Controlled double-slit electron diffraction,” New Journal of Physics 15, 033018; author-institution repository.
- Feynman, Leighton and Sands, The Feynman Lectures on Physics, III.1.
- What counts as an observer?: detection, retention and interpretation.