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Which Causality? Differences between Trajectory and Copenhagen Analyses of an Impulsive Perturbation

Edward R. Floyd

Published Jan 1, 1998• 18 citations
OpenAlex

Abstract

The trajectory and Copenhagen representations render different predictions for impulsive perturbations. The different predictions are due to the different roles that causality plays in the trajectory and Copenhagen interpretations. We investigate a small perturbing impulse acting on the ground state of an infinitely deep square well. For the two representations, the first-order perturbation calculations for the temporal change in energy differ. This temporal change in energy for the trajectory representation is dependent upon the microstate of the wave function. We show that even under Copenhagen epistemology, the two representations predict different theoretical results. PACS Numbers: 3.65.Bz; 3.65.Ca

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The trajectory and Copenhagen representations render different predictions for impulsive perturbations. The different predictions are due to the different roles that causality plays in the trajectory and Copenhagen interpretations. We investigate a small perturbing impulse acting on the ground state of an infinitely deep square well. For the two representations, the first-order perturbation calculations for the temporal change in energy differ. This temporal change in energy for the trajectory representation is dependent upon the microstate of the wave function. We show that even under Copenhagen epistemology, the two representations predict different theoretical results. PACS Numbers: 3.65.Bz; 3.65.Ca
The trajectory and Copenhagen representations render different predictions for impulsive perturbations. The different predictions are due to the different roles that causality plays in the trajectory and Copenhagen interpretations. We investigate a small perturbing impulse acting on the ground state of an infinitely deep square well. For the two representations, the first-order perturbation calculations for the temporal change in energy differ. This temporal change in energy for the trajectory representation is dependent upon the microstate of the wave function. We show that even under Copenhagen epistemology, the two representations predict different theoretical results.
We introduce the de Broglie-Bohm causal interpreation of quantum mechanics and compare it to the standard interpretation of quantum mechanics, the Copenhagen interpretation. We examine the possibility of experimentally distinguishing between the two theories, as well as the potential for the causal interpretation to more easily bridge the gap between the physics of the quantum and classical worlds. We then use the causal interpretation to construct a deterministic model of the helium atom in which the two electrons move along trajectories through space and time about a stationary nucleus. The dynamics are governed by the non-relativistic Schrödinger equation and the spin vectors of both electrons are assumed to be constant along their respective trajectories. We examine the Bohmian trajectories associated with (approximations to) eigenstates of the helium Hamiltonian as well as the trajectories associated with some non-eigenstates. We also compute an approximation to the ground state energy of the helium atom using a representation of the helium wavefunction in terms of hydrogenic eigenfunctions which is motivated by a perturbation approach.
The trajectory representation in the classical limit (¯h → 0) manifests a residual indeterminacy. We show that the trajectory representation in the classical limit goes to neither classical mechanics (Planck’s correspondence principle) nor statistical mechanics. This residual indeterminacy is contrasted to Heisenberg uncertainty. We discuss the relationship between residual indeterminacy and ’t Hooft’s information loss and equivalence classes. PACS Numbers: 3.65.Bz; 3.65.Ca Key words: classical limits, trajectory interpretation, Planck’s correspondence principle, residual indeterminacy, ’t Hooft’s
Quantum mechanics describes the behavior of the quantum objects, but the description and the interpretation of its basic principles; (nature of the wave, interpretations of: superposition principle, complementary principle, and uncertainly principle), these latter have had many debates concerning their meanings and their concepts between the scientists. Since\nthe beginning of its appearance until today, many different interpretations between many schools and scientists have emerged, about the descriptions of quantum mechanics. In our present contribution, a modest modeling for many worlds interpretation is presented, followed by solving and interpreting a famous example “particle in an infinite square potential\nwell”, using our modeling and then, a comparison between Copenhagen and many world interpretations.