The relationship between conservation of energy and conservation of probability.
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If you shine light into a contraption that causes interference, the total brightness (energy) you get out equals the energy you put in. Fire one photon into the same contraption, and it ends up somewhere, with probabilities proportional to brightness from the first experiment.
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The time-dependent Schrödinger equation says that the time-derivative of state is given by an imaginary multiple of an operator that you get by contracting an energy observable with (the contravariant version of) an inner product representing probability.
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Observable and inner product both mean symmetric sesquilinear forms, and the Schrödinger equation treats the energy observable and the probability inner product almost symmetrically, so it implies conservation of energy and of probability for essentially the same reasons.
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