Double-Slit Interference Pattern for a Macroscopic Quantum System

Abstract

In this study, we solve analytically the Schrödinger equation for a macroscopic quantum oscillator as a central system coupled to two environmental micro-oscillating particles. Then, the double-slit interference patterns are investigated in two limiting cases, considering the limits of uncertainty in the position probability distribution. Moreover, we analyze the interference patterns based on a recent proposal called stochastic electrodynamics with spin. Our results show that when the quantum character of the macro-system is decreased, the diffraction pattern becomes more similar to a classical one. We also show that, depending on the size of the slits, the predictions of quantum approach could be apparently different with those of the aforementioned stochastic description.

Langevin equation for a dissipative macroscopic quantum system: Bohmian theory versus quantum mechanics

“…where T (t − t 0 ) was defined in (22). For ω c (t − t 0 ) 1 and considering J(ω α ) in (20), the retarded resistance function T (t − t 0 ) approaches zero, as is obvious in (23).…”

“…where α = 1 to N . So, the wave function of the ground state of the Hamiltonian (8) is obtained as [20,21] ψ 0 (q,…”

“…where the prime inṼ q(t) and f α q(t) denotes q−differentiation. Now, we define Ohmic distribution for the frequencies of the environmental harmonic oscillators ω e as J(ω e ) λ Q ω e e −ωe/ωc (20) where λ Q is a positive constant which quantifies the strength of the interaction between the environment and the macro-system and ω c is the cut-off frequency which maximizes J(ω e ). By using the bilinear model, after some calculations, one can derive the equation of motion of the central system as…”

“…is the memory term. For the temporal domain 1 ω c (t − t 0 ) and using Markov approximation, the memory term for Ohmic frequency distribution defined in (20) becomes M (t) λ Qq (t) [8]. In other words, the nature of the dissipation is contained in the memory function.…”

“…is the force applied on the central system at t > 0. For a MQS having a nearly sharp distribution of position q (see, e.g., [20]), the final form of the quantum langevin equation becomesq…”