3 - The probability interpretation

Summary

Historical remarks

The quantum mechanical formalism discovered by Heisenberg [Heis 25] an Schrödinger [Schrö 26] in 1925 was first interpreted in a statistical sense by Born [Born 26]. The formal expressions p(φ,a_i) = |(φ,φ^ai)|², i ∈ N, were interpreted as the probabilities that a quantum system S with preparation φ possesses the value a_i that belongs to the state φ^ai. This original Born interpretation, which was formulated for scattering processes, was, however, not tenable in the general case. The probabilities must not be related to the system S in state φ, since in the preparation φ the value a_i of an observable A is in general not subjectively unknown but objectively undecided. Instead, one has to interpret the formal expressions p(a_i, a_i) as the probabilities of finding the value a_i after measurement of the observable A of the system S with preparation φ. In this improved version, the statistical or Born interpretation is used in the present-day literature.

On the one hand, the statistical (Born) interpretation of quantum mechanics is usually taken for granted, and the formalism of quantum mechanics is considered as a theory that provides statistical predictions referring to a sufficiently large ensemble of identically prepared systems S(φ) after the measurement of the observable in question. On the other hand, the meaning of the same formal terms p(φ,a_i) for an individual system is highly problematic.