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,ai) = |(φ,φai)|2, i ∈ N, were interpreted as the probabilities that a quantum system S with preparation φ possesses the value ai that belongs to the state φai. 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 ai of an observable A is in general not subjectively unknown but objectively undecided. Instead, one has to interpret the formal expressions p(ai, ai) as the probabilities of finding the value ai 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,ai) for an individual system is highly problematic.