# Hydrodynamical derivation of liquid He4 energy spectrum (but really it's the math)

1. Apr 24, 2009

### SilverHawk

1. The problem statement, all variables and given/known data

I'm following an article by Pitaevskii from 1956, and there's one mathematical transition which I don't understand. The Hamiltonian is given first in coordinate space, and then the density operator is transformed to its Fourier components. The continuity equation is also used, and then I haven't been able to get to the expression obtained in the article for the Hamiltonian.

2. Relevant equations

Attached as word document.

3. The attempt at a solution

I basically tried to "reverse engineer" my way from the equation I'm trying to obtain to the Hamiltonian in Fourier representation; namely, to express the density Fourier components using the current density Fourier components (according to the Fourier continuity equation), and then to plug those into the expression I'm trying to obtain for the Hamiltonian. The problem is that the term with the cross product (see doc) isn't zero, and I don't see how it would cancel out. And otherwise I don't see how the equation is obtained, and I highly doubt that Pitaevskii got it wrong.

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