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I have a dilute weakly-interacting bose gas and make the assumption that I have only s-wave scattering. Then I'm able to write the Hamiltonian as:

$$ H=\sum\frac{p^2}{2m}a_p^{\dagger}a_p+\frac{1}{2V}V_0\sum a_{p_1+q}^{\dagger}a_{p_2-q}^{\dagger}a_{p_1}a_{p_2} $$

Now, I can use Bogoliubov's approach and replace $a_0$ with a c-number. When I do this to second order, I will receive the Bogoliubov dispersion law.

My question is now, how does this refer to the Gross-Pitaevskii equation? What are the differences and similarities? Do they both belong togehter?

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    $\begingroup$ For an interesting and lucid review of the relationship between these two approximations, see Leggett, New J. Phys. 5, 103 (2003). $\endgroup$ – Mark Mitchison Oct 28 '15 at 17:16
  • $\begingroup$ Thank you! I will have a look on this one. Since it's from Leggett, it will be probable useful. :-) $\endgroup$ – QuantumMechanics Oct 29 '15 at 14:38
  • $\begingroup$ I would be also happy about hearing some intuitions :-) $\endgroup$ – QuantumMechanics Oct 29 '15 at 15:33
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    $\begingroup$ The Gross-Pitaevskii's equation is obtained from the Heisenberg equations-of-motion for the normal and anomalous bosonic Green's functions in the ordered (superfluid) phase. For a detailed account, see Chp. 14, section 55 of Fetter, A. L., and J. D. Walecka. “Quantum theory of many-particle systems” (1971). $\endgroup$ – AlQuemist Dec 5 '15 at 15:41

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