Momentum Space

Yajiang Hao, Yiwang Liu, Xiangguo Yin

In momentum space, we investigate the correlation properties of the ground state of Tonks–Girardeau gases. With Bose–Fermi mapping method, the exact ground state wavefunction in coordinate space can be obtained based on the wavefunction of spin-polarized Fermions. By Fourier transformation we obtain the ground state wavefunction in momentum space, and therefore the pair correlation and the reduced one-body density matrix (ROBDM) in momentum space, whose diagonal part is the momentum distribution. The ROBDM in momentum space is the Fourier transformation of the ROBDM in coordinate space and the pair correlation in momentum space is the Fourier transformation of the reduced two-body density matrix in coordinate space. The correlations in momentum space display larger values only in small momentum region and vanish in most other regions. The lowest natural orbital and occupation distribution in momentum space are also obtained.

Harry A. Mavromatis

Sergei P. Efimov

The fundamental quantum Coulomb problem in the momentum space is considered. A differential equation with SO(4) symmetry has been obtained in the momentum space instead of the integral Fock equation. The corresponding equation in the coordinate space is the sum of the squares of the angular momentum and Runge–Lenz operators. This approach is unknown in the momentum space where the Runge–Lenz operator is not applied. The Runge–Lenz operator obtained in the momentum space is simpler than that in the coordinate space and allows one to effectively consider the Coulomb problem in the momentum space. A relation of new operator to the infinitesimal rotation operator of the three-dimensional Fock sphere has been determined.

T.T. TAYLOR

Sergei P. Efimov

The fundamental quantum Coulomb problem in the momentum space is considered. A differential equation with SO(4) symmetry has been obtained in the momentum space instead of the integral Fock equation. The corresponding equation in the coordinate space is the sum of the squares of the angular momentum and Runge–Lenz operators. This approach is unknown in the momentum space where the Runge–Lenz operator is not applied. The Runge–Lenz operator obtained in the momentum space is simpler than that in the coordinate space and allows one to effectively consider the Coulomb problem in the momentum space. A relation of new operator to the infinitesimal rotation operator of the three-dimensional Fock sphere has been determined.