Renninger negative-result experiment

In quantum mechanics, the Renninger negative-result experiment is a thought experiment that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics. The statement is that a particle need not be detected in order for a quantum measurement to occur, and that the lack of a particle detection can also constitute a measurement. The thought experiment was first posed in 1953 by Mauritius Renninger. It can be understood to be a refinement of the paradox presented in the Mott problem.

The Mott problem concerns the paradox of reconciling the spherical wave function describing the emission of an alpha ray by a radioactive nucleus, with the linear tracks seen in a cloud chamber. Formulated in 1929 by Sir Nevill Francis Mott and Werner Heisenberg, it was resolved by a calculation done by Mott that showed that the correct quantum mechanical system must include the wave functions for the atoms in the cloud chamber as well as that for the alpha ray. The calculation showed that the resulting probability is non-zero only on straight lines raying out from the decayed atom; that is, once the measurement is performed, the wave-function becomes non-vanishing only near the classical trajectory of a particle.

In the Renninger formulation, the cloud chamber is replaced by a pair of hemispherical particle detectors, completely surrounding a radioactive atom at the center that is about to decay by emitting an alpha ray. For the purposes of the thought experiment, the detectors are assumed to be 100% efficient, so that the emitted alpha ray is always detected.

By consideration of the normal process of quantum measurement, it is clear that if one detector registers the decay, then the other will not: a single particle cannot be detected by both detectors. The core observation is that the non-observation of a particle on one of the shells is just as good a measurement as detecting it on the other.

The strength of the paradox can be heightened by considering the two hemispheres to be of different diameters; with the outer shell a good distance farther away. In this case, after the non-observation of the alpha ray on the inner shell, one is led to conclude that the (originally spherical) wave function has "collapsed" to a hemisphere shape, and (because the outer shell is distant) is still in the process of propagating to the outer shell, where it is guaranteed to eventually be detected.

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