Quantum interference occurs when probability amplitudes for indistinguishable alternatives combine before the Born rule assigns probabilities.
A superposition of two distinct energy eigenstates gives the expectation value of a time-independent observable an interference term proportional to . The displayed beat period applies when that term has nonzero amplitude. A vanishing off-diagonal matrix element, zero state coefficient or equal energies instead gives a constant expectation value.
Two indistinguishable bosons entering separate inputs of an ideal balanced beam splitter have zero probability of leaving one per output. The two alternatives for that occupancy interfere destructively. Distinguishability in temporal, spatial or internal modes weakens the cancellation.
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