The attached presentation is the second one of a sequence of three that we want to do on Quantum Mechanics using Computer Algebra. The first presentation was about the equation for a quantum system of identical particles, the GrossPitaevskii equation (GPE). This second presentation is about the spectrum of its solutions. The level is that of an advanced undergraduate QM course. The novelty is again in the way these problems can be tackled nowadays in a computer algebra worksheet with Physics.
The GrossPitaevskii equation and Bogoliubov spectrum
Pascal Szriftgiser^{1} and Edgardo S. ChebTerrab^{2}
(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F59655, France
(2) Maplesoft, Canada
Departing from the equation for a quantum system of identical boson particles, i.e.the GrossPitaevskii equation (GPE), the dispersion relation for planewave solutions are derived, as well as the Bogoliubov equations and dispersion relations for small perturbations around the GPE stationary solutions.

Stationary and planewave solutions to the GrossPitaevskii equation


Problem: Given the GrossPitaevskii equation,

b) Derive the dispersion relation for planewave solutions as a function of G and .

Background: The GrossPitaevskii equation is particularly useful to describe Bose Einstein condensates (BEC) of cold atomic gases [3, 4, 5]. The temperature of these cold atomic gases is typically in the w100 nanoKelvin range. The atomatom interaction are repulsive for and attractive for , where G is the interaction constant. The GPE is also widely used in nonlinear optics to model the propagation of light in optical fibers.


The Bogoliubov equations and dispersion relations


Problem: Given the GrossPitaevskii equation,

b) Show that the dispersion relations of these equations, known as the Bogoliubov spectrum, are given by


,


References
[1] GrossPitaevskii equation (wiki)
[2] Continuity equation (wiki) [3] Bose–Einstein condensate (wiki)
[4] Dispersion relations (wiki)
[5] Advances In Atomic Physics: An Overview, Claude CohenTannoudji and David GueryOdelin, World Scientific (2011), ISBN10: 9812774963.
[6] Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN13: 9780123970237.

QuantumMechanics.pdf Download QuantumMechanics2.mw
Edgardo S. ChebTerrab
Physics, Maplesoft
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