Modified Schrodinger-Poisson equation: Quantum polytropes

Choptuik, Matthew W.; Shinkaruk, David

doi:10.1103/physrevd.96.103010

Cited by 2 publications

(1 citation statement)

References 16 publications

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“…The importance of a nonlinear superposition property for the log SE is already under investigation [26]. Solving a logarithmic version of the Schrödinger-Poisson equation could even be contemplated to further cosmological investigations into dark matter [27,28].…”

Section: Resultsmentioning

confidence: 99%

Solution of the 3D logarithmic Schrödinger equation with a central potential

Shertzer

Scott

2020

J. Phys. Commun.

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Some form of the time-independent logarithmic Schrödinger equation (log SE) arises in almost every branch of physics. Nevertheless, little progress has been made in obtaining analytical or numerical solutions due to the nonlinearity of the logarithmic term in the Hamiltonian. Even for a central potential, the Hamiltonian does not commute with L 2 or L ; z the Hamiltonian is invariant under the parity operation only if the wave function is an eigenstate of the parity operator. We show that the solutions with well-defined parity can be expressed as a linear combination of eigenstates of L 2 and L , z where the parity restrictions on l and m determine the nodal structure of the wave function. The dominant contribution in the sum is designated as l˜and m; these serve as approximate quantum numbers. Using an iterative finite element approach, we also carry out fully converged numerical calculations in 1D, 2D and 3D for the special case of a Coulomb potential. The nodal structure of the wave functions and the expectation values á ñ L 2 and á ñ L z 2 are consistent with the analytical predictions. Values for the energy and expectations values are tabulated for the low-lying states. The methods developed for this problem are applicable across many areas of physics.

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Section: Resultsmentioning

confidence: 99%

Solution of the 3D logarithmic Schrödinger equation with a central potential

Shertzer

Scott

2020

J. Phys. Commun.

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Approximate Solution of Coupled Schrödinger and Poisson Equation in Inversion Layer Problem: An Approach Based on Homotopy Perturbations

Kevkić¹,

Stojanović²

2019

Zeitschrift Für Naturforschung A

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In this paper, the homotopy perturbation method (HPM) is applied to the coupled set of Schrödinger–Poisson (SP) equations in inversion layer problem for obtaining the approximate analytical solution. Inversion layer of n-type is considered, and the electric quantum limit is assumed. By introducing some dimensionless quantities, the SP system has been turned into one which can be solved along the infinite interval. After some appropriate transformations, the infinite interval has been reduced to finite one $(0,1)$, and recurrence series of the HPM approximate solutions of the coupled SP system have been obtained. The existence and convergence of obtained HPM approximate solutions have been formally proved. Moreover, these solutions show relative simple mathematical form, as well as high degree of accuracy what is desirable for semiconductor device modelling.

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Modified Schrodinger-Poisson equation: Quantum polytropes

Cited by 2 publications

References 16 publications

Solution of the 3D logarithmic Schrödinger equation with a central potential

Solution of the 3D logarithmic Schrödinger equation with a central potential

Approximate Solution of Coupled Schrödinger and Poisson Equation in Inversion Layer Problem: An Approach Based on Homotopy Perturbations

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