Possible implications of exponential decay

• Published in: Annals of physics Vol. 217; no. 2; pp. 222 - 278
• Main Authors: Steyerl, A, Malik, S.S
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.08.1992; Elsevier
• Subjects: 661100 - Classical & Quantum Mechanics- (1992-); 662200 - Specific Theories & Interaction Models; CENTER-OF-MASS SYSTEM; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Classical and quantum physics: mechanics and fields; COHERENT SCATTERING; DIFFERENTIAL EQUATIONS; DIFFRACTION; ENERGY; ENTROPY; EQUATIONS; EQUATIONS OF MOTION; Exact sciences and technology; FREE ENERGY; MECHANICS; Pa; PARTIAL DIFFERENTIAL EQUATIONS; PARTICLE KINEMATICS; Particle Systematics- (1992-); PARTICLES; PHYSICAL PROPERTIES; Physics; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; PROBABILITY; QUANTUM MECHANICS; RANDOMNESS; SCATTERING; SCHROEDINGER EQUATION; SEMICLASSICAL APPROXIMATION; STATISTICAL MECHANICS; THERMODYNAMIC PROPERTIES; UNCERTAINTY PRINCIPLE; WAVE EQUATIONS; Equation of motion; Particle motion; Wave function; Quantum statistics; Entropy; Free energy; Schrödinger equation; Lifetime; Numerical solution; Center of mass; Particle system; Nonlinearity; Unstable system; Wave mechanics
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/0003-4916(92)90152-C

Semiclassical concepts are developed which could make the appearance of a logarithmic nonlinearity in a Schrödinger-type equation plausible. Extending previous work by Białynicki-Birula and Mycielski (BBM), this approach is based on the introduction of a novel wave function describing the center of mass (CM) motion of unstable particles or composite systems subject to statistical changes of their internal quantum state. The element of statistical randomness associated with a purely exponential decay law (if that exists) suggests the use of thermodynamic concepts like entropy and free energy. The entropy is associated with the probability of existence of the state in time and space. These concepts are applied only to a domain “blurred” by the quantum uncertainty principle where the problematic definition of a time and entropy operator might be possible. The paper consists of three main parts. Section 1 develops an extended nonrelativistic equation of motion. The proposed equation contains yet reinterprets the BBM equation, and for stable systems it reduces to the Schrödinger equation. Definite predictions are made of observable quantities. For the free neutron, the prediction for the amplitude b of the logarithmic term is 3.7 × 10 −19 eV. In Section 2, the family of localized, nonspreading ground-state solutions to the BBM equation (the “gaussons”) is extended, in two and three spatial dimensions, to states classified by finite quantized angular momenta and definite values of entropy. The statistical behavior of CM systems and their electromagnetic interaction are investigated. In Section 3, implications of these concepts are outlined with emphasis on possible experimental manifestations. Suggested laboratory tests include high-precision measurements of unstable particle diffraction on linear gratings as well as neutron interferometer experiments of the type previously attempted to test the BBM equation. An improvement in resolution by several orders of magnitude is required. A further testing possibility is the investigation of particle resonances. An important feature of the present model is a subtle combination of quantum and classical aspects, achieved without compromising fundamental principles (like quantum statistical properties), while reinterpreting microreversibility. For stable systems the proposed model identically reduces to standard wave mechanics.