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Physical Review A

Kinetic-energy density functional for a special shape-invariant potential of a one-dimensional two-level system

Jiushu Shao, John J. Stezowski

Published Nov 1, 1990• 3 citations
Publisher

Abstract

By introducing the concept of supersymmetry to quantum mechanics, several authors have shown that exactly soluble potentials, including all those well known, are easily constructed. Is the kinetic-energy density functional corresponding to these potentials a simple form? We show that the answer is no, even to the simplest one, the harmonic potential, if one builds the kinetic-energy density functional from the reduced density matrix.

DOI
10.1103/physreva.42.5767

Related papers

Reply to ‘‘Comment on ‘Kinetic-energy density functional for a special shape-invariant potential of a one-dimensional two-level system’ ’’

Jiushu Shao, John J. Stezowski

For a harmonic potential, which is the special shape-invariant potential discussed in our paper [1], the relation between the ground state and the first-excited-state wave functions [Eq. (2)] should be . . . Therefore, we have shown that one cannot find a closed form of the kinetic-energy density for the harmonic oscillator.

Physical Review A•Sep 1, 1991•1 citation

Comment on ‘‘Kinetic-energy density functional for a special shape-invariant potential of a one-dimensional two-level system’’

Jiqiang Chen, Malcolm J. Stott

This Comment points out misleading aspects of a recent paper by Shao and Stezowski [Phys. Rev. A 42, 5767 (1990)].Received 19 December 1990DOI:https://doi.org/10.1103/PhysRevA.44.3387©1991 American Physical Society

Physical Review A•Sep 1, 1991•2 citations

Exact single-particle kinetic energy functional for general two-level and modeln-level one-dimensional systems: Dependence only on electron density and its gradient

Pawel Michal Kozlowski, N. H. March

From the explicit solution of a two-level problem previously obtained, it is proved that the single-particle kinetic energy density functional ${t}_{s}$[\ensuremath{\rho}] is the density \ensuremath{\rho} times a function solely of the ratio \ensuremath{\rho}'/\ensuremath{\rho}, with \ensuremath{\rho}' the first derivative of \ensuremath{\rho}. For an n-level system with a harmonic-oscillator potential, the form of ${t}_{s}$[\ensuremath{\rho}] is explicitly obtained. Again ${t}_{s}$[\ensuremath{\rho}] involves only \ensuremath{\rho} and \ensuremath{\rho}'; no higher derivatives appear.

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Single-particle kinetic energy in density-functional theory: Harmonic confinement in two and three dimensions

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A Study about the Supersymmetry in the context of Quantum Mechanics

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Dirac density matrix and the Legendre transform of the kinetic energy generated by one-dimensional model potentials

I. A. Howard, Norman Henry March

In density functional theory, the single-particle kinetic energy is still not known in an orbital-free form. The natural tool to calculate such kinetic energy for a given one-body potential V is the Dirac density matrix y. Here, by first taking solvable one-dimensional examples, and in particular the sech 2 (x) potential, forms for the Dirac density matrix are exhibited in terms of the potential.This in turn generates the Legendre transform of the single-particle kinetic energy. Finally, one-dimensional perturbation theory for the kinetic energy density, summed to all orders in V, is presented in the appendix.

Journal of Physics A Mathematical and General•Dec 21, 2005•0 citations
By introducing the concept of supersymmetry to quantum mechanics, several authors have shown that exactly soluble potentials, including all those well known, are easily constructed. Is the kinetic-energy density functional corresponding to these potentials a simple form? We show that the answer is no, even to the simplest one, the harmonic potential, if one builds the kinetic-energy density functional from the reduced density matrix.
From the explicit solution of a two-level problem previously obtained, it is proved that the single-particle kinetic energy density functional ts{t}_{s}ts​[\ensuremath{\rho}] is the density \ensuremath{\rho} times a function solely of the ratio \ensuremath{\rho}'/\ensuremath{\rho}, with \ensuremath{\rho}' the first derivative of \ensuremath{\rho}. For an n-level system with a harmonic-oscillator potential, the form of ts{t}_{s}ts​[\ensuremath{\rho}] is explicitly obtained. Again ts{t}_{s}ts​[\ensuremath{\rho}] involves only \ensuremath{\rho} and \ensuremath{\rho}'; no higher derivatives appear.
Using the known perturbation series for the idempotent Dirac density matrix in powers of a given one-body potential V(r)V(\mathbf{r})V(r), Stoddart and March (SM) generated a corresponding series for the kinetic energy density. While the general term of the SM series is known, a summation has not been achieved to date. A contribution to solve this problem is made here by exhibiting an explicit form of the above kinetic energy density for Fermions filling an arbitrary number of closed shells, when the confinement is harmonic. This example is of considerable current interest because of ongoing experiments on ultracold atomic gases of Fermions.
In this work we present an introduction to Supersymmetry in the context of 1-dimensional Quantum Mechanics. For that purpose we develop the concept of hamiltonians factorization using the simple harmonic oscillator as an example, we introduce the supersymmetric oscilator and, next, we generalize these concepts to introduce the fundamentals of Supersymmetric Quantum Mechanics. We also discuss useful tools to solve problems in Quantum Mechanics which are intrinsecally related to Supersymmetry as hierarchy of hamiltonians and shape invariance. We present two approximation methods which will be specially useful: the well known Variational Method and the Logarithmic Perturbation Theory, the latter being closely related to the concepts of superpotentials and hierarchy of hamiltonians. Finally, we present problems related to superpotentials which are monomials in even powers of the x coordinate multiplied by the sign function epsilon(x), which seems to be a new class of problems in Supersymmetric Quantum Mechanics.
In density functional theory, the single-particle kinetic energy is still not known in an orbital-free form. The natural tool to calculate such kinetic energy for a given one-body potential V is the Dirac density matrix y. Here, by first taking solvable one-dimensional examples, and in particular the sech 2 (x) potential, forms for the Dirac density matrix are exhibited in terms of the potential.This in turn generates the Legendre transform of the single-particle kinetic energy. Finally, one-dimensional perturbation theory for the kinetic energy density, summed to all orders in V, is presented in the appendix.