Analyzing the Time Evolution of a Particle by Decomposes the Initial State Confinement in 1D Well into the Lowest Eigenstates Energy

Authors

DOI:

https://doi.org/10.55544/jrasb.3.2.17

Keywords:

Schrodinger-equation, Particle-in-Box, Confinement, Time-evolution

Abstract

In this work, we obtained the time evolution of the wave function of a limited quantum system (1D Box), hence getting a mathematical model to describe the system. By using programming computes, it performs a time evolution that decomposes the initial state into the 2,10, and 20 lowest energy eigenstates. Finally, by comparing numerical de-composition coefficients for the wave function to the analytical values, it found the number of knots increases directly versus the energy of the particle's quantum state. As a result, the mean bending given by the second derivative which is proportional to the kinetic energy operator should increase. We found there is a negligible mean and standard deviation of the energy in units of the ground state energy.

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References

Roy A. K. (2015). Quantum confinement in 1D systems through an imaginary-time evolution method. Modern Physics Letters A Vol 30 Issue (37), pp.1550176.‏ https://doi:10.1142/S021773231550176X

Aharonov Y., Anandan J., & Vaidman L. (1993). Meaning of the wave function, Physical Review A Vol 47(6), pp. 4616.‏ https://doi.org/10.1103/PhysRevA.47.4616

Fano G., & Blinder S. M. (2017). Twenty-first century quantum mechanics: Hilbert space to quantum computers. Springer International Publishing. Doi: 10.1007/978-3-319-58732-5.

Roy A. K., Gupta N., & Deb B. M. (2001). Time-dependent quantum-mechanical calculation of ground and excited states of anharmonic and double well oscillators. Physical Review A Vol 65 Issue (1). pp. 012109.‏ https://doi.org/10.1103/PhysRevA.65.012109

Michels A., De Boer J., & Bijl A. (1937). Remarks concerning molecular interaction and their influence on the polarizability. Physica, Vol. 4 Issue (10). pp. 981-994.‏ https://doi.org/10.1016/S0031-8914(37)80196-2.

Sommerfeld A., & Welker H. (1938). Künstliche Grenzbedingungen beim Keplerproblem. Annalen der Physik Vol. 424 Issue (1‐2) pp. 56-65.‏ https://doi.org/10.1002/andp.19384240109.

Bertini B., Heidrich-Meisner F., Karrasch C., Prosen T., Steinigeweg R., & Žnidarič M. (2021). Finite-temperature transport in one-dimensional quantum lattice models. Reviews of Modern Physics Vol. 93 Issue (2). pp.025003.‏ https://doi.org/10.1103/RevModPhys.93.025003.

Stenzel O. (2022). The Schrödinger Equation and Model System I, In Light–Matter Interaction. Springer Cham.‏ pp.55-77. Doi: 10.1007/978-3-030-87144-4.

Nandy D. K., & Sowiński T. (2020). Dynamical properties of a few mass-imbalanced ultracold fermions confined in a double-well potential. New Journal of Physics Vol. 22 Issue (5). pp.053043. https://doi.org/10.1088/1367-2630/ab878c .

Nandy D. K., & Sowiński T. (2021). Dynamical resistivity of a few interacting fermions to the time-dependent potential barrier. New Journal of Physics Vol.23 Issue (4). PP.043019.‏ https://doi.org/10.48550/arXiv.2101.00892

Pomorski K., Giounanlis P., Blokhina E., Leipold D., Pęczkowski P., & Staszewski R. B. (2019). From two types of electrostatic position-dependent semiconductor qubits to quantum universal gates and hybrid semiconductor-superconducting quantum computer. In Superconductivity and Particle Accelerators Vol. 11054. pp.147-166.‏ doi:10.1117/12.2525217.

Juang C., Kuhn K. J., & Darling R. B. (1990). Stark shift and field-induced tunneling in Al x Ga 1− x As/GaAs quantum-well structures. Physical Review B Vol. 41 Issue (17). pp.12047.‏ https://doi.org/10.1103/PhysRevB.41.12047.

Bader P., Blanes S., & Casas F. (2013). Solving the Schrödinger eigenvalue problem by the imaginary time propagation technique using splitting methods with complex coefficients. The Journal of chemical physics Vol. 139Issue (12). pp.124117.‏ https://doi.org/10.1063/1.4821126.

Jørgensen L., Cardozo D. L., & Thibierge E. (2011). Numerical Resolution of The Schrödinger Equation. Tech. Rep. (École Normale Supérieure de Lyon).

Lehtovaara L., Toivanen J., & Eloranta J. (2007). Solution of time-independent Schrödinger equation by the imaginary time propagation method. Journal of Computational Physics Vol.221 Issue (1). pp.148-157.‏ https://doi.org/10.1016/j.jcp.2006.06.006.

Strickland M., & Yager-Elorriaga D. (2010). A parallel algorithm for solving the 3d Schrödinger equation. Journal of Computational Physics Vol.229 Issue (17). pp.6015-6026.‏ DOI: 10.1016/j.jcp.2010.04.032

Chin S. A., Janecek S., & Krotscheck E. (2009). Any order imaginary time propagation method for solving the Schrödinger equation. Chemical Physics Letters Vol.470 Issue (4-6). pp, 342-346.‏ DOI: 10.1016/j.cplett.2009.01.068

Arteca G. A., Maluendes S. A., Fernández F. M., & Castro E. A. (1983). Discussion of several analytical approximate expressions for the eigenvalues of the bounded harmonic oscillator and hydrogen atom. International journal of quantum chemistry Vol.24 Issue (2). pp.169-184.‏ https://doi.org/10.1002/qua.560240205

Vawter R., (1968). Effects of finite boundaries on a one-dimensional harmonic oscillator. Physical Review Vol.174 Issue (3). pp.749.‏ https://doi.org/10.1103/PhysRev.174.749.

Breinig M., (2009). Wave Mechanics. In Compendium of Quantum Physics Springer. Berlin Heidelberg.‏ pp.822-827. Doi:10.1007/978-3-540-70626-7_231.

Amorós Trepat M., (2021). Quantum particles in fractal external potential. (Bachelor's thesis, Universitat Politècnica de Catalunya)‏. http://hdl.handle.net/2117/356702.

Sehra A. S. (2007). Finite element analysis of the Schrödinger equation. arXiv preprint arXiv: 0704.3240.‏ https://doi.org/10.48550/arXiv.0704.3240.

Rieth M., Schommers W., & Baskoutas S. (2002). Exact numerical solution of Schrödinger's equation for a particle in an interaction potential of general shape. International Journal of Modern Physics B Vol.16 Issue (27). pp.4081-4092.‏ doi/abs/10.1142/S0217979202014802.

Jaschke D., Wall M. L., & Carr L. D. (2018). Opensource matrix product states: Opening ways to simulate entangled many-body quantum systems in one dimension. Computer Physics Communications Vol.225. pp.59-91. https://doi.org/10.1016/j.cpc.2017.12.015‏

Kunstatter G., & Das S. (2022). A First Course on Symmetry, Special Relativity and Quantum Mechanics: The Foundations of Physics. Springer.‏‏ pp.213-252.

Sudiarta I. W., & Geldart D. W. (2009). The finite difference time domain method for computing the single-particle density matrix. Journal of Physics A: Mathematical and Theoretical Vol.42 Issue (28). pp.285002. ‏Doi: 10.1088/1751-8113/42/28/285002

Kosugi T., Nishiya Y., & Matsushita Y. I. (2021).‏ Probabilistic imaginary-time evolution by using forward and backward real-time evolution with a single ancilla: first-quantized eigen solver of quantum chemistry for ground states, arXiv preprint arXiv:2111.12471.

Ting-Yun S., Cheng-Guang B. A. O., & Bai-Wen L. I. (2001). Energy spectra of the confined atoms obtained by using B-splines, Communications in Theoretical Physics. Vol.35 Issue (2) pp.195.‏ DOI: 10.1088/0253-6102/35/2/195

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Published

2024-04-13

How to Cite

Oglah, M. H. (2024). Analyzing the Time Evolution of a Particle by Decomposes the Initial State Confinement in 1D Well into the Lowest Eigenstates Energy. Journal for Research in Applied Sciences and Biotechnology, 3(2), 103–107. https://doi.org/10.55544/jrasb.3.2.17

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