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Enter Poem or Quote (Required)Required y Schrödinger equation, mean-field approaches are commonly developed. A detailed derivation can be found in Dalfovo’s article. The generalization of the original Bogoliubov description to the physical situation in real experiments is given by describing the field operators in the Heisenberg: (2) Where y (x,t) is a complex function defined as the expectation value of the field operator and its modulus represents the condensate density. If we ignored the depletion of the condensate, the time evolution of the condensate wave function at temperature T=0 is obtained by taking the ansatz for the field operator and using the Heisenberg equation. Then we can get the Gross-Pitaevskii equation for the mean field (see some condensed matter physics text books such as by Chaikin), (3) The description of the propagation of noninteracting matter waves in periodic potentials is straightforward once one has found the eigenstates and corresponding eigenenergies of the system. If we consider a one dimensional sinusoidal periodic potential of the form,
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