Comments on Ziman's Quantum Hydrodynamics
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概要
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Ziman has tried to guantize the motion of the fluid including vortex motion, and found that the Hamiltonian consists of three parts. i. e. the phonon part, the roton part and the remsining part, and obtained E<SUB>r</SUB>=Δ<SUB>0</SUB>+h<SUP>2</SUP>k<SUP>2</SUP>/2m<SUP>*</SUP> for the roton spectrum. On the other hand, it has been noticed by Kaempffer that the remaining part, which contains the phonon-roton interaction, phonon-phonon interaction and other higher-order processes, can not be treated as a small perturbation and the roton spectrum obtained by Ziman might have to be modified. In this paper, we transform Ziman's Hamiltonian into the appropriate form which consists of new three parts, i. e. the new phonon part, the new roton part and the remaining part which can be considered as a small perturbation. Thus we have the energy spectrum of the form E<SUB>r</SUB>=Δ+h<SUP>2</SUP>k<SUP>2</SUP>/2μ for the "noton" spectrum, where Δ and μ are different from Ziman's values. It is also noted that the nomenclature "roton" should not necessarily correspond to the quantized vortex motion, because Δ in the roton spectrum tends to zero in the limit of incompressional fluid.
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