Math Porn By the Mossad
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,
Copyright © Leroy Baker | Year Posted 2021
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