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Journal of the Physical Society of Japan

The Angular Momentum Paradox of3He-A

Takafumi Kita

Published Mar 15, 1996• 18 citations
Publisher

Abstract

The kinetic energy of a charged system is calculated for Mermin's one-particle density matrix of Fermi superfluids. The final expression consists of two separate contributions from the center-of-mass and the relative motions of an effective bound-state wave function. The relative part naturally includes coupling of the intrinsic orbital angular momentum density to the magnetic field. For 3 He- A at T =0, its magnitude is \((\hbar / 2) \times \)(number density), in agreement with Ishikawa's result.

DOI
10.1143/jpsj.65.664

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The kinetic energy of a charged system is calculated for Mermin's one-particle density matrix of Fermi superfluids. The final expression consists of two separate contributions from the center-of-mass and the relative motions of an effective bound-state wave function. The relative part naturally includes coupling of the intrinsic orbital angular momentum density to the magnetic field. For 3 He- A at T =0, its magnitude is ((\hbar / 2) \times )(number density), in agreement with Ishikawa's result.
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We study the effects of twisting on the low-energy excitations of superfluid 3A^{3}\mathit{A}3A. They are small compared to the effects of bending, but they have a different symmetry which makes them important. The structure of the excitation wave functions has strong analogies with the eigenstates of a charged particle in a magnetic field. The group velocity is parallel to curl l^. The density of states at zero energy has a contribution proportional to twisting which is odd with respect to k^. Because of this term a density fluctuation produces a fluctuation of the current carried by the excitations. This current fluctuation turns out to be exactly equal to the fluctuation of the C0{C}_{0}C0​ term in the current. This shows that the C0{C}_{0}C0​ term fluctuation is carried by the normal liquid and leads us to believe that the C0{C}_{0}C0​ term as a whole is linked to the normal liquid. Since calculations making use of the Bardeen-Cooper-Schrieffer wave function consider only the condensate, our results offer a possible explanation for the ``angular-momentum paradox.'' We physically interpret the vn{\mathrm{v}}_{n}vn​ term in the Josephson equation at T=0 and show that it is readily obtained from the microscopic description of the normal liquid. In the same way we deduce from the momentum conservation law an equation of motion for vn{\mathrm{v}}_{n}vn​ and we show that it can be entirely understood from the microscopic framework.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
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