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## Quantum Mechanics using Computer Algebra

The attached presentation is the first one of a sequence of three that we wanted to do on Quantum Mechanics using Computer Algebra. The level is that of an advanced undergraduate QM course. Tackling this topic within a computer algebra worksheet in the way it's done below, however, is an entire novelty, and illustrates well the kind of computations that can be done today with Maple & Physics.

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

(2) Maplesoft

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 [6].

 Solution
 Continuity equation for a quantum system of identical particles

References

[4] Bose-Einstein Condensation in Dilute Gases, C. J. Pethick and H. Smith, Second Edition, Cambridge (2008), ISBN-13: 978-0521846516.

[5] Advances In Atomic Physics: An Overview, Claude Cohen-Tannoudji and David Guery-Odelin, World Scientific (2011), ISBN-10: 9812774963.

[6] Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN-13: 978-0123970237.