Unlocking a 1956 Article: Pitaevskii's Hamiltonian Transformation

[SilverHawk]

Homework Statement

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.

Homework Equations

Attached as word document.

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.

 

[Jhero]

Thank you for sharing your thoughts and attempts at solving this problem. It seems like you have put a lot of effort into understanding the mathematical transition in Pitaevskii's article. I agree that it can be frustrating when a certain step in a mathematical derivation is not clear.

One approach you could try is to break down the problem into smaller steps and see if you can understand each step individually before putting them together. For example, you could try to understand the Fourier transformation of the density operator first, and then move on to the continuity equation. Once you have a good understanding of each individual step, it may be easier to see how they all fit together to obtain the final expression for the Hamiltonian.

Also, keep in mind that mathematical derivations can sometimes involve multiple steps and simplifications, so it's possible that Pitaevskii may have used some intermediate steps or approximations that are not explicitly mentioned in the article. It may be helpful to consult other sources or textbooks on the subject to see if they offer any insights or alternative approaches.

Overall, don't get discouraged and keep exploring different avenues to try and understand the problem. it's important to be persistent and curious in the face of challenging problems. Good luck!