Ground state of a quantum system of identical boson particles
Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2
(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France
Departing from the Energy of a quantum system of identical boson particles, the field equation is derived. This is the Gross-Pitaevskii equation (GPE). A continuity equation for this system is also derived, showing that the velocity flow satisfies , i.e.: is irrotational.
The Gross-Pitaevskii equation
Problem: derive the field equation describing the ground state of a quantum system of identical particles (bosons), that is, the Gross-Pitaevskii equation (GPE).
Background: The Gross-Pitaevskii equation is particularly useful to describe Bose Einstein condensates (BEC) of cold atomic gases [3, 4, 5], that is, an ensemble of identical quantum boson particles that interact with each other with an interaction constant G. The temperature of these cold atomic gases is typically in the w100 nano-Kelvin range. The atom-atom interactions are repulsive for and attractive for (which could lead to some instabilities). The GPE is also widely used in non-linear optics to model the propagation of light in optical fibers. In this area, GPE is known as "non-linear Schrödinger equation", and the non-linearity comes from the Kerr effect .
 Gross-Pitaevskii equation (wiki)
 Continuity equation (wiki)
 Bose–Einstein condensate (wiki)
 Bose-Einstein Condensation in Dilute Gases, C. J. Pethick and H. Smith, Second Edition, Cambridge (2008), ISBN-13: 978-0521846516.
 Advances In Atomic Physics: An Overview, Claude Cohen-Tannoudji and David Guery-Odelin, World Scientific (2011), ISBN-10: 9812774963.
 Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN-13: 978-0123970237.