Numerical solution of one‐dimensional time‐independent Schrödinger equation by using symplectic schemes

Liu, Xue‐Shen; Liu, Xiaoyan; Zhou, Zhóngyuan; Ding, Pei; Shou-Fu, Pan

doi:10.1002/1097-461x(2000)79:6<343::aid-qua2>3.3.co;2-f

Cited by 11 publications

(16 citation statements)

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“…We consider the one-dimensional eigenvalue problem with boundary conditions ψ(a) = 0, ψ(b) = 0 [5]. We use the shooting scheme in the implementation of the above methods.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…These are methods that preserve the area and the volume in phase space due to the symplectic properties. It was pointed out originally by Liu et al [5] that symplectic integrators are suitable methods for the numerical integration of the Schrödinger equation.…”

Section: Methodsmentioning

confidence: 99%

“…In the literature many numerical methods have been developed to solve the time-independent Schrödinger equation [14,15]. Symplectic integrators are suitable methods for the numerical solution of the Schrödinger equation [5], among their properties is the energy preservation, which is an important property in quantum mechanics. In this Letter we construct a new third order Symplectic Partitioned Runge-Kutta method (SPRK) and a modification of this method with the trigonometrically fitted property.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge–Kutta methods

Monovasilis¹,

Kalogiratou²,

Simos

2008

Physics Letters A

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“…We consider the one-dimensional eigenvalue problem with boundary conditions ψ(a) = 0, ψ(b) = 0 [5]. We use the shooting scheme in the implementation of the above methods.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge–Kutta methods

Monovasilis¹,

Kalogiratou²,

Simos

2008

Physics Letters A

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“…when E ∈ [−50, 0]) we have the well-known bound-states problem while in the case of positive eigenenergies (i.e. when E ∈ (0, 1000]) we have the well-known resonance problem (see [119][120][121][122]128,129,132,[134][135][136][137]). Many problems in chemistry, physics, physical chemistry, chemical physics, electronics etc., are expressed by Eq.…”

Section: Numerical Examplementioning

confidence: 99%

High order closed Newton-Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation

Simos

2012

J Math Chem

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In this paper we study the connection between: (i) closed Newton-Cotes formulae of high order, (ii) trigonometrically-fitted and exponentially-fitted differential methods, (iii) symplectic integrators. Several one step symplectic integrators have been produced based on symplectic geometry during the last decades (see the relevant literature and the references here). However, the study of multistep symplectic integrators is very poor. In this paper we investigate the High Order Closed Newton-Cotes Formulae and we write them as symplectic multilayer structures. We develop trigonometrically-fitted and exponentially-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the symplectic schemes in order to solve the resonance problem of the radial Schrödinger equation. Based on the theoretical and numerical results, conclusions on the efficiency of the new obtained methods are given.Highly Cited

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“…The Morse potential is of the form Now we employ numerical schemes M1-M6 with the symplectic scheme-shooting method [19] and the bisection method [20] to compute the energy eigenvalues E n of the time-independent Schrödinger equation. Boundary conditions are q min = 0 and q max = 0, where we take min = −15.5 and max = 15.5.…”

Section: Morse Potentialmentioning

confidence: 99%

Optimized third-order force-gradient symplectic algorithms

2010

Sci. China Phys. Mech. Astron.

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With the natural splitting of a Hamiltonian system into kinetic energy and potential energy, we construct two new optimal thirdorder force-gradient symplectic algorithms in each of which the norm of fourth-order truncation errors is minimized. They are both not explicitly superior to their no-optimal counterparts in the numerical stability and the topology structure-preserving, but they are in the accuracy of energy on classical problems and in one of the energy eigenvalues for one-dimensional time-independent Schrödinger equations. In particular, they are much better than the optimal third-order non-gradient symplectic method. They also have an advantage over the fourth-order non-gradient symplectic integrator. symplectic integrators, symplectic scheme-shooting method, celestial mechanics, time-independent Schrödinger equation, energy eigenvalues, numerical stability, bisection method, topological structure PACS: 02.70.-c, 03.65.-w, 95.10.Ce

show abstract

Numerical solution of one‐dimensional time‐independent Schrödinger equation by using symplectic schemes

Cited by 11 publications

References 0 publications

Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge–Kutta methods

Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge–Kutta methods

High order closed Newton-Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation

Optimized third-order force-gradient symplectic algorithms

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