Quantum Indeterminacy - Measurement

Measurement

An adequate account of quantum indeterminacy requires a theory of measurement. Many theories have been proposed since the beginning of quantum mechanics and quantum measurement continues to be an active research area in both theoretical and experimental physics. Possibly the first systematic attempt at a mathematical theory was developed by John von Neumann. The kind of measurements he investigated are now called projective measurements. That theory was based in turn on the theory of projection-valued measures for self-adjoint operators which had been recently developed (by von Neumann and independently by Marshall Stone) and the Hilbert space formulation of quantum mechanics (attributed by von Neumann to Paul Dirac).

In this formulation, the state of a physical system corresponds to a vector of length 1 in a Hilbert space H over the complex numbers. An observable is represented by a self-adjoint (i.e. Hermitian) operator A on H. If H is finite dimensional, by the spectral theorem, A has an orthonormal basis of eigenvectors. If the system is in state ψ, then immediately after measurement the system will occupy a state which is an eigenvector e of A and the observed value λ will be the corresponding eigenvalue of the equation A e = λ e. It is immediate from this that measurement in general will be non-deterministic. Quantum mechanics, moreover, gives a recipe for computing a probability distribution Pr on the possible outcomes given the initial system state is ψ. The probability is

where E(λ) is the projection onto the space of eigenvectors of A with eigenvalue λ.