Eigenvalue problem of the Schrödinger equation via the finite-difference time-domain method

A method is presented to calculate the eigenenergies and eigenfunctions of quantum wires. This is a true three-dimensional method based on a direct implementation of the time-dependent Schrödinger equation. It makes no approximations to the Schrödinger equation other than the finite-difference approximation of the space and time derivatives. The accuracy of our method is tested by comparing it to analytical results in a cylindrical wire.

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“…We begin describing the FDTD implementation of the time-dependent Schrödinger equation [ 13 , 14 ], which is written in the following form …”

Section: The Finite-difference Time-domain Methodsmentioning

confidence: 99%

Time-Domain Simulation of Three Dimensional Quantum Wires

2016

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“…The proposed procedures which is based on the BOR-FDTD method and the Wick rotation transformation has several advantages over the traditional 3D-FDTD method [33]. First, due to employing only real-valued wavefunctions rather than dealing with complex wavefunctions and having two coupled Schrödinger equations, the simulation needs less running-time and memory.…”

Section: Q-bor-fdtd Algorithm For Three-dimensional Schrödinger Equationmentioning

confidence: 99%

“…The FDTD method [26] is a numerical technique that has been used in several branches of Physics. It has been extensively employed in various applications in electrodynamics, nano-optics, and nanophotonics for solving Maxwell's equations and also in quantum mechanics for solving the Schrödinger equation [27][28][29][30][31][32][33][34][35][36][37]. This kind of the FDTD application has been established by Sullivan et al (2001), and Soriano et al (2004), to study the eigenvalues, eigenstates and dynamics of several quantum nanostructures [35], such as, quantum well wires [34], spin evolution and two electrons in a quantum dot [36,37].…”

Section: Introductionmentioning

confidence: 99%

Q-BOR–FDTD method for solving Schrödinger equation for rotationally symmetric nanostructures with hydrogenic impurity

Firoozi¹,

Mohammadi²,

Khordad³

et al. 2022

Phys. Scr.

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An efficient method inspired by the traditional body of revolution finite-difference time-domain (BOR-FDTD) method is developed to solve the Schrodinger equation for rotationally symmetric problems. As test cases, spherical, cylindrical, cone-like quantum dots, harmonic oscillator, and spherical quantum dot with hydrogenic impurity are investigated to check the efficiency of the proposed method which we coin as Quantum BOR-FDTD (Q-BOR-FDTD) method. The obtained results are analysed and compared to the 3-D FDTD method, and the analytical solutions. Q-BOR-FDTD method proves to be very accurate and time and memory efficient by reducing a three-dimensional problem to a two-dimensional one, therefore one can employ very fine meshes to get very precise results. Moreover, it can be exploited to solve problems including hydrogenic impurities which is not an easy task in the traditional FDTD calculation due to singularity problem. To demonstrate its accuracy, we consider spherical and cone-like core-shell QD with hydrogenic impurity. Comparison with analytical solutions confirms that Q-BOR–FDTD method is very efficient and accurate for solving Schrodinger equation for problems with hydrogenic impurity

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“…An FDTD method for electronic systems has already been proposed [12][13][14], called the FDTD method for quantum device (FDTD-Q). In the FDTD-Q method, electronic wave function is discretized and divided into real and imaginary parts, ψ(r)=ψ R (r)+iψ I (r), and then the Schrödinger equation is solved by temporally and alternately calculating ψ R (t) and ψ I (r), in a similar manner to the FDTD method.…”

Section: Introductionmentioning

confidence: 99%

Finite-difference time-domain method for calculating photoexcited molecular vibration wave-packet dynamics

Oka

2018

J. Phys. Commun.

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We propose a simple numerical calculation method for solving molecular vibrational dynamics. A finite-difference time-domain (FDTD) method is utilized to solve the Schrödinger equation for wave packet dynamics driven by incident light pulses. Vibrational wave function is discretized and divided into real and imaginary parts, and then the Schrödinger equation is solved by temporally and alternately calculating the real and imaginary parts, in a similar manner to the conventional FDTD method. Taking generation of a hot molecule by pump-dump scheme and multi-photon excitation process as examples, we show the validity and usefulness of the proposed method. The dynamics of the vibrational wave packets in the pump-dump scheme and transient virtual excitations of intermediate vibronic state in the three-photon absorption can be easily visualized, and the intensity dependence of excitation population for ultra-high intensity beyond the perturbation theory can be accurately calculated.

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Eigenvalue problem of the Schrödinger equation via the finite-difference time-domain method

Cited by 14 publications

References 19 publications

Time-Domain Simulation of Three Dimensional Quantum Wires

Time-Domain Simulation of Three Dimensional Quantum Wires

Q-BOR–FDTD method for solving Schrödinger equation for rotationally symmetric nanostructures with hydrogenic impurity

Finite-difference time-domain method for calculating photoexcited molecular vibration wave-packet dynamics

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