The Ensemble Interpretation of Probability

Thomas A. Brody

Paul M. Sheldon

A density operator, $ρ= {P}_{α} |α> = 0$, ${P}_{α}$ and ${P}_{β}$ cannot be interpreted as probabilities of finding $|α>$ and $|β>$ respectively. We show that a density matrix comprised of two (${P}_{α}$ and ${P}_{β}$ nonzero) non-orthogonal projectors have unique spectral decomposition into diagonal form with orthogonal projectors. Only then, according to axioms of Von Neumann and Fock, can we have probability interpretation of that density matrix, only then can the diagonal elements be interpreted as probabilities of an ensemble. Those probabilities on the diagonal are not ${P}_{α}$ and ${P}_β$. Further, only in the case of orthogonal projectors can we have the degenerate situation in which multiple ensembles are permitted.

Ciann‐Dong Yang, Chia-Hung Wei

The current trajectory interpretation of quantum mechanics is based on an ensemble viewpoint that the evolution of an ensemble of Bohmian trajectories guided by the same wavefunction Ψ converges asymptotically to the quantum probability . Instead of the Bohm's ensemble-trajectory interpretation, the present paper gives a single-trajectory interpretation of quantum mechanics by showing that the distribution of a single chaotic complex-valued trajectory is enough to synthesize the quantum probability. A chaotic complex-valued trajectory manifests both space-filling (ergodic) and ensemble features. The space-filling feature endows a chaotic trajectory with an invariant statistical distribution, while the ensemble feature enables a complex-valued trajectory to envelop the motion of an ensemble of real trajectories. The comparison between complex-valued and real-valued Bohmian trajectories shows that without the participation of its imaginary part, a single real-valued trajectory loses the ensemble information contained in the wavefunction Ψ, and this explains the reason why we have to employ an ensemble of real-valued Bohmian trajectories to recover the quantum probability . © 2015 Wiley Periodicals, Inc.

Jagdish Mehra, Helmut Rechenberg

김재영

I examined the Statistical Ensemble Interpretation (SEI) of quantum mechanics. Historical reappraisal of Max Born's statistical interpretation shows precisely how Born used the concept of the probability. After investigating the so-called Ensemble Interpretation proposed by Albert Einstein based on ...

Peter J. Clark