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 Gross-Pitaevskii 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 Gross-Pitaevskii equation and Bogoliubov spectrum

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft, Canada


Departing from the equation for a quantum system of identical boson particles, i.e.the Gross-Pitaevskii equation (GPE), the dispersion relation for plane-wave solutions are derived, as well as the Bogoliubov equations and dispersion relations for small perturbations `δϕ` around the GPE stationary solutions.

Stationary and plane-wave solutions to the Gross-Pitaevskii equation


Problem: Given the Gross-Pitaevskii equation, NULL

I*`ℏ`*psi[t] = (G*abs(psi)^2+V)*psi-`ℏ`^2*%Laplacian(psi)/(2*m)


a) Derive a relationship between the chemical potential mu entering the phase of stationary, uniform solutions, the atom-atom interaction constant G and the particle density n = abs(psi)^2 in the absence of an external field (V = 0).


b) Derive the dispersion relation for plane-wave solutions as a function of G and n. 



Background: The Gross-Pitaevskii 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 nano-Kelvin range. The atom-atom interaction are repulsive for G > 0 and attractive for G < 0 , where G is the interaction constant. The GPE is also widely used in non-linear optics to model the propagation of light in optical fibers.



The Bogoliubov equations and dispersion relations



Problem: Given the Gross-Pitaevskii equation,


a) Derive the Bogoliubov equations, that is, equations for elementary excitations `&delta;&varphi;` and conjugate(`&delta;&varphi;`)around a GPE stationary solution `&varphi;`(x, y, z), NULL


"{[[i `&hbar;` (&PartialD;)/(&PartialD;t) `delta&varphi;`=-(`&hbar;`^2 (&nabla;)^2`delta&varphi;`)/(2 m)+(2 G |`&varphi;`|^2+V-mu) `delta&varphi;`+G `&varphi;`^2 (`delta&varphi;`),,],[i `&hbar;` (&PartialD;)/(&PartialD;t)( `delta&varphi;`)=+(`&hbar;`^2 (&nabla;)^2(`delta&varphi;`))/(2 m)-(2 G |`&varphi;`|^2+V-mu) (`delta&varphi;`)-G `delta&varphi;` ((`&varphi;`))^(2),,]]"


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


epsilon[k] = `&hbar;`*omega[k] and `&hbar;`*omega[k] = `&+-`(sqrt(`&hbar;`^4*k^4/(4*m^2)+`&hbar;`^2*k^2*G*n/m)),


where k is the wave number of the considered elementary excitation, epsilon[k] its energy or, equivalently, omega[k] its frequency.






[1] Gross-Pitaevskii equation (wiki)

[2] Continuity equation (wiki)
[3] Bose–Einstein condensate (wiki)

[4] Dispersion relations (wiki)

[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.



QuantumMechanics.pdf     Download

Edgardo S. Cheb-Terrab
Physics, Maplesoft

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