The energy spectrum of the excitations in liquid helium

by Cohen, Michael

Abstract (Summary)
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Feynman has used the wave function [...] to represent an excitation (phonon or roton) of momentum [...] in liquid helium. [...] is the ground state wave function, and the sum runs over all the atoms in the liquid. The resultant energy spectrum is correct for phonons [...] and has the qualitatively correct feature of exhibiting a minimum at [...] (roton region). Landau and subsequent workers have shown that the specific heat and second sound velocity data require the value [...], where [...] is the minimum roton energy and [...] is Boltzmann's constant. Feynman's energy spectrum locates the minimum correctly but gives [...]. A wave function of the form [...] is proposed here to represent an excitation of momentum [...]. The function g represents the fact that the neighbors of a moving atom execute some smooth pattern of backflow around it; g is taken as the potential function for a dipole velocity field, the strength of the dipole being left arbitrary until the end of the computation. To facilitate computation, it proves useful to replace [...]. This procedure is mathematically legitimate, not only because [...] is small, but because the wave function is inserted into a variational principle for the energy and is guaranteed to yield an overestimate. The strength of the dipole is finally chosen to minimize the energy yielding the new value [...]. The optimal value for the dipole strength is very close to the "classical" value which one would expect on the basis of a current conservation argument.
Bibliographical Information:

Advisor:R.P. Feynman

School:California Institute of Technology

School Location:USA - California

Source Type:Master's Thesis



Date of Publication:01/01/1956

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