Background: For a BEC close to its ground state (at temperature T = 0 K), its excitations are well described by small perturbations around the stationary state of the BEC. The energy of an excitation is then given by the Bogoliubov dispersion relation (derived previously in Mapleprimes "Quantum Mechanics using computer algebra II").
where G is the atomatom interaction constant, n is the density of particles, m is the mass of the condensed particles, k is the wavevector of the excitations and their pulsation ( time the frequency). Typically, there are two possible types of excitations, depending on the wavevector :
Problem: An impurity of mass moves with velocity within such a condensate and creates an excitation with wavevector . After the interaction process, the impurity is scattered with velocity .
a) Departing from Bogoliubov's dispersion relation, plus energy and momentum conservation, show that, in order to create an excitation, the impurity must move with an initial velocity

When , no excitation can be created and the impurity moves through the medium without dissipation, as if the viscosity is 0, characterizing a superfluid. This is the Landau criterion for superfluidity.

b) Show that when the atomatom interaction constant (repulsive interactions), this value is equal to the group velocity of the excitation (speed of sound in a condensate).