Exact canonically conjugate momenta approach to a one-dimensional neutron–proton system, I

Abstract

Introducing collective variables, a collective description of nuclear surface oscillations has been developed with the first quantized language, contrary to the second quantized one in Sunakawa's approach for a Bose system. It overcomes difficulties remaining in the traditional theories of nuclear collective motions: Collective momenta are not exact canonically conjugate to collective coordinates and are not independent. On the contrary to such a description, Tomonaga first gave the basic idea to approach elementary excitations in a one-dimensional Fermi system. The Sunakawa's approach for a Fermi system is also expected to work well for such a problem. In this paper, on the isospin space, we define a density operator and further following Tomonaga, introduce a collective momentum. We propose an exact canonically momenta approach to a one-dimensional neutron-proton (N-P) system under the use of the Grassmann variables.

Velocity operator approach to a Fermion system

“…. Such an evaluation is quite different from the manner adopted in the preceding papers [4,5] in which only the term Λ 1(i) k had been taken into account. As a result, the present evaluation leads to another approximate formula for f (i) (ρ; k) and another final form of [v k and T 0 (ρ), thus, the correct expression for T 0 (ρ) could be determined in terms of ρ k and the constant C 0 appearing in the expansion of a modified T 0 (ρ) turns out to involve the term <ρ k ρ −k > Ave which is not determined yet.…”

“…N Ω ν(k) . Further we derive the Bogoliubov transformation for Boson-like operators θa k and a † k θ which already appeared in the preceding papers [4,5] and is the same form as the famous Bogoliubov transformation for the usual Bosons [7] but which brings the modified single-particle energy √ 7 2 ε k in the quasi-particle excitation energy E k in the lowest order approximation. Thus, we could provide the modified and then the correct theory for the velocity operator approach.…”

“…which already appeared in the preceding papers [4,5] and is of the same form as the Bogoliubov transformation for the usual Bosons [6] except the introduction of the Grassmann numbers θ and θ. The above kind of the diagonalization (3.27) has also been given by Sunakawa in the Boson system [11].…”

“…On the other hand, Sunakawa's discrete integral equation method for a Fermi system [3] may be expected to also work well for a collective motion problem. In the preceding papers [4,5], introducing density operators ρ k and associated variables π k and defining exact momenta Π k (collective variables), we could get an exact canonically conjugate momenta approach to one-and three-dimensional (1D and 3D) Fermion systems. In the present paper, we formulate a velocity operator approach to a 3D Fermion system.…”

A possible quantum fluid-dynamical approach to vortex motion in nuclei

“…The Sunakawa's method, which is applicable also to a Fermi system, may work well for such a problem. It has been considered in the exact canonical momenta approach to a neutron-proton system [14].…”