Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 96

1  1 Time evolution 1-a Functional variation 1-b Variational computation: the time-dependent Gross-Pitaevskii equation 1-c Phonons and Bogolubov spectrum

2  2 Hydrodynamic analogy 2-a Probability current 2-b Velocity evolution

3  3 Metastable currents, superfluidity 3-a Toroidal geometry, quantization of the circulation, vortex 3-b Repulsive potential barrier between states of different l 3-c Critical velocity, metastable flow 3-d Generalization; topological aspects

In this complement, we return to the calculations of Complement C_XV, concerning a system of bosons all in the same individual state. We now consider the more general case where that state is time-dependent. Using a variational method similar to the one we used in Complement C_XV, we shall study the time variations of the N-particle state vector. This amounts to using a time-dependent mean field approximation. We shall establish in § 1 a time-dependent version of the Gross-Pitaevskii equation, and explore some of its predictions such as the small oscillations associated with Bogolubov phonons. In § 2, we shall study local conservation laws derived from this equation for which we will give a hydrodynamic analogy, introducing a characteristic relaxation length. Finally, we will show in § 3 how the Gross-Pitaevskii equation predicts the existence of metastable flows and superfluidity.