6 - Bohr's theory of measurement

Summary

I propose now to examine Bohr's informal theory of measurement from a somewhat more formal point of view, and to consider how his theory relates to the orthodox theory of measurement in quantum mechanics. This is a difficult task, since Bohr nowhere discusses the theory of measurement in formal terms. My discussion, therefore, will be less closely tied to Bohr's own statements. I shall also consider some important objections to the theory.

The objective-values theory of measurement

Before we look into this theory, it will be useful to have a brief sketch of some of the relevant formal elements of quantum mechanics.

In the Hilbert space formulation of quantum mechanics the measurable physical properties (‘observables’) of an object are represented by linear Hermitian operators A, B, … which have complete orthonormal sets of eigenvalues a_n, b_n, … corresponding to the subspaces (closed linear manifolds) of a Hilbert space H. These operators are in one-to-one correspondence with the projection measures P^A, P^B, … on the Hilbert space such that P^A = Σ_na_nP_n, where P_n are the projections of the eigenvectors ψ_n belonging to the eigenvalues a_n of A, and a_n are discrete, non-degenerate values (for simplicity I shall deal only with operators which have such values, and shall denote observables and the corresponding operators by the same symbols).