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PSA Proceedings of the Biennial Meeting of the Philosophy of Science Association

What Was Born's Statistical Interpretation?

Linda Wessels

Published Jan 1, 1980• 35 citations
Publisher

Abstract

When Max Born introduced his statistical interpretation of wave mechanics in mid-1926, the reactions were mixed. His most recent collaborator, Pascual Jordan, was delighted; his former assistant, Werner Heisenberg, was horrified. The interpretation served as grist for Neils Bohr's philosophical mill. For Bohr's former collaborator, John Slater, it inspired a strange hybrid interpretation, a sort of statistical version of Schrödinger's electrodynamic model. What was the interpretation that engendered these varied reactions? Neither Born nor most of his contemporaries saw in it anything that even suggests the interpretation that we now associate with his name. Rather, it was another former assistant of Born's, Wolfgang Pauli, with eyes as sharp as his tongue, who saw in Born's work the idea that |Ψ(q)| represents the probability distribution for position.

DOI
10.1086/psaprocbienmeetp.1980.2.192590

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When Max Born introduced his statistical interpretation of wave mechanics in mid-1926, the reactions were mixed. His most recent collaborator, Pascual Jordan, was delighted; his former assistant, Werner Heisenberg, was horrified. The interpretation served as grist for Neils Bohr's philosophical mill. For Bohr's former collaborator, John Slater, it inspired a strange hybrid interpretation, a sort of statistical version of Schrödinger's electrodynamic model. What was the interpretation that engendered these varied reactions? Neither Born nor most of his contemporaries saw in it anything that even suggests the interpretation that we now associate with his name. Rather, it was another former assistant of Born's, Wolfgang Pauli, with eyes as sharp as his tongue, who saw in Born's work the idea that |Ψ(q)| represents the probability distribution for position.
Abstract This chapter covers the emergence of the statistical interpretation of the new quantum mechanics. After covering the earliest suggestions in this direction by Born and Pauli, it covers the statistical transformation theory of Dirac and Jordan, focusing on Jordan’s version. This version was an important part of the inspiration of Heisenberg’s uncertainty paper, which is covered next. The chapter concludes with the Como and Solvay conferences of late 1927 where Bohr revealed his complementarity principle and where the Copenhagen-Göttingen matrix-mechanics camp of Bohr, Born, Heisenberg, and Pauli squared off with Schrödinger, Einstein, De Broglie, and others in the wave-mechanics camp over the interpretation of the new quantum formalism.
Schrödinger hoped that his wave mechanics would help to re-establish some sense of ‘visualizability’ of the physics going on inside the atom. In searching for a suitable interpretation of the wavefunction, he focused on the density of electrical charge, which he associated with the wavefunction ψ‎ multiplied by its complex conjugate. Hidden in his words is the interpretation that would eventually come to dominate our understanding of the wavefunction. Max Born had no hesitation in concluding that the only way to reconcile wave mechanics with the particle description is to interpret the modulus-square of the wavefunction as a probability density. It was Wolfgang Pauli who proposed to interpret this not only as a transition probability or as the probability for the system to be in a specific state, as Born had done, but as the probability of ‘finding’ the electron at a specific position in its orbit inside an atom.
Position probabilities play a privileged role in the interpretation of quantum mechanics. The quantum wave function (usually designated “Ψ(r)’) cannot itself represent a physical quantity, because it is complex valued. Its square — |Ψ(r)|2 — however, is real; and it is |Ψ(r)|2 which is the fundamental interpreted quantity of the theory. Schrödinger, who originated wave mechanics, interpreted |Ψ(r)|2 as the density of a charged particle; but this suggestion was inadequate. A principal reason is that, for most physical systems, |Ψ(r)|2 spreads out in time, whereas particles themselves do not. Schrödinger's interpretation was replaced by Born who introduced probabilities into quantum mechanics. Born urged that |Ψ(r)|2 represents not the density of the system in space, but rather a probability density. This is almost universally how quantum mechanics is taught today: |Ψ(r)|2 gives the probability that a system in state Ψ(r) is located at r.
At the 1927 Solvay conference Born and Heisenberg presented a joint report on quantum mechanics. I suggest that the significance of this report lies in that it contains a ‘final’ formulation of the statistical interpretation of quantum mechanics that goes beyond Born’s origi-nal proposal. In particular, this formulation imports elements from Heisenberg’s work as well as from the transformation theory of Dirac and Jordan. I suggest further a reading of Born and Heisenberg’s position in which the wave function is an effective notion. This can make sense of a remarkable aspect of their presentation, namely the fact that the ‘quantum mechanics’ of Born and Heisenberg apparently lacks wave function collapse.
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.