Ensembles in Quantum Mechanics

Abstract

The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which immediately become unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.-Albert Einstein (1879-1955) [1], p. 671. This quotation is perhaps the most famous statement of the ensemble interpretation of quantum mechanics. The role of the ensemble in quantum mechanics ranges from innocuous to profound, and even controversial. The innocuous role of the ensemble stems from the fact that quantum mechanics does not predict the actual events, but only the probabilities of the various possible outcomes (probability in quantum mechanics) of the various possible events. In order to compare the predictions of quantum mechanics with experiment, one must prepare a state and measure some dynamical variable, and repeat this preparation-measurement sequence many times. The relative frequencies of the various outcomes in this ensemble of results can then be compared with the theoretical probabilities predicted by quantum mechanics. Thus it is natural to say that quantum mechanics describes the statistics of an ensemble of similarly prepared systems. Here, as in classical statistical mechanics, one should not confuse the ensemble of systems with an assembly of systems into a composite. For example, if the system is a single particle, then the ensemble is a conceptual set of replicas of it, each in its own environment, whereas the assembly would be a many-particle system. The role of the ensemble is to enable statistical analysis; its members do not interact with or influence each other. The more significant role of the ensemble interpretation is exemplified by Schrödinger's cat paradox [2], which involves an unstable atom, a cat, and a mechanism that releases a poison to kill the cat when the atom decays. The initial state vector of the system, |φ 1 |live, describes an atom in an excited state and a live cat. The final state vector, after the atom has decayed and the cat is dead, will