QM-Simulator-2D

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QM-Simulator-2D

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Interactive simulation of a quantum particle in 2D.

QM-Simulator-2D 최신버젼 다운로드

최종 버전 다운로드 (.zip)

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# Quantum Mechanics in 2D
This [WebGL program](https://marl0ny.github.io/QM-Simulator-2D/index.html) simulates the quantum mechanics of a single particle confined in a 2D box, where inside this box the user can create new potential barriers and scatter Gaussian wave packets off them. The full instructions are found [here](https://github.com/marl0ny/QM-Simulator-2D/blob/main/INSTRUCTIONS.md).

The default integration method used in this program is described in page 690 of An Introduction to Computer Simulation Methods by H. Gould et al (which references an [article](https://aip.scitation.org/doi/pdf/10.1063/1.168415) by P. Visscher). This method basically involves splitting the complex-valued wave function into its real and imaginary components, where each component is updated separately using the other for each time step. Other integration methods are available as well, such as the [Crank-Nicolson](https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method) and [Split-Operator](https://www.algorithm-archive.org/contents/split-operator_method/split-operator_method.html) method. For more information on how to use these various methods in the simulation, refer to the "Integration Methods" section of the [instructions](https://github.com/marl0ny/QM-Simulator-2D/blob/main/INSTRUCTIONS.md) page.

A few examples are showcased in this [YouTube playlist](https://www.youtube.com/playlist?list=PLLMjc99uROx53gM9lZIDJNM1OrlfuTJ2P).

#### Dirac simulation (work in progress)

Also included in this repository is a [simulation of a 2D relativistic quantum particle](https://marl0ny.github.io/QM-Simulator-2D/rel-qm.html) that uses the Dirac instead of the Schrödinger equation. The Dirac equation is solved numerically by placing each of the two two-component spinors that constitute the four-component Dirac wave function along separate staggered spatial grids, where each spinor is updated using the other at each step in time. This [method is from an article](https://arxiv.org/abs/1306.5895) by R. Hammer and W. Pötz.

[A separate Dirac simulation that uses the Split-Operator method](https://marl0ny.github.io/QM-Simulator-2D/rel-qm-split-step.html) instead of the one above is included as well.

## APIs and Frameworks Used
- The [WebGL](https://www.khronos.org/webgl/) API
- [dat.gui](https://github.com/dataarts/dat.gui) by [Google Data Arts Team](https://github.com/dataarts)
- [stats.js](https://github.com/mrdoob/stats.js/) by the stats.js authors
- [jszip](https://stuk.github.io/jszip/) by [Stuart Knightley](https://github.com/Stuk), [David Duponchel](https://github.com/dduponchel), Franz Buchinger, António Afonso

# References
 - Gould, H., Tobochnik J., Christian W. (2007). Quantum Systems.
 In An Introduction to Computer Simulation Methods, 
chapter 16. Pearson Addison-Wesley.

- Visscher, P. (1991). A fast explicit algorithm for the time‐dependent Schrödinger equation. Computers in Physics, 5, 596-598. [https://doi.org/10.1063/1.168415](https://doi.org/10.1063/1.168415)
 
 - Schroeder D. [Quantum Scattering in Two Dimensions](https://physics.weber.edu/schroeder/software/QuantumScattering2D.html).

Crank-Nicolson Method:

- Wikipedia contributors. (2021, October 6). [Crank-Nicolson method](https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method). In Wikipedia, The Free Encyclopedia.

- Wikipedia contributors. (2021, August 1). [Jacobi method](https://en.wikipedia.org/wiki/Jacobi_method). In Wikipedia, The Free Encyclopedia.

- Sadovskyy I., Koshelev A., Phillips C., Karpeyev D., Glatz A. (2015). Stable large-scale solver for Ginzburg-Landau equations for superconductors. Journal of Computational Physics 294, 639-654. [https://doi.org/10.1016/j.jcp.2015.04.002](https://doi.org/10.1016/j.jcp.2015.04.002)

Split-Operator Method:

- James Schloss. [The Split-Operator Method](https://www.algorithm-archive.org/contents/split-operator_method/split-operator_method.html). In The Arcane Algorithm Archive.

- Wikipedia contributors. (2021, May 6). [Split-step method](https://en.wikipedia.org/wiki/Split-step_method). In Wikipedia, The Free Encyclopedia.

- Bauke, H., Keitel, C. (2011). Accelerating the Fourier split operator method via graphics processing units. Computer Physics Communications, 182(12), 2454-2463. [https://doi.org/10.1016/j.cpc.2011.07.003](https://doi.org/10.1016/j.cpc.2011.07.003)

Dirac Equation:

- Wikipedia contributors. (2021, June 16). [Dirac equation](https://en.wikipedia.org/wiki/Dirac_equation). In Wikipedia, The Free Encyclopedia.

- Wikipedia contributors. (2021, August 5). [Dirac spinor](https://en.wikipedia.org/wiki/Dirac_spinor). In Wikipedia, The Free Encyclopedia.

- Hammer, R., Pötz W. (2014). Staggered grid leap-frog scheme for the (2 + 1)D Dirac equation. Computer Physics Communications, 185(1), 40-53. [https://doi.org/10.1016/j.cpc.2013.08.013](https://doi.org/10.1016/j.cpc.2013.08.013)

Nonlinear Schrödinger Equation:

- Antoine, X., Bao, W., Besse C. (2013). Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations. Computer Physics Communications, 184(12), 2621-2633. [https://doi.org/10.1016/j.cpc.2013.07.012](https://doi.org/10.1016/j.cpc.2013.07.012)

- Ira Moxley III, F. (2013). [Generalized finite-difference time-domain schemes for solving nonlinear Schrödinger equations](https://digitalcommons.latech.edu/cgi/viewcontent.cgi?article=1284&context=dissertations). Dissertation, 290.

###

Laplacian Stencils:

- Wikipedia contributors. (2021, February 17)
 [Discrete Laplacian Operator 1.5.1 Implementation via operator discretization](https://en.wikipedia.org/wiki/Discrete_Laplace_operator#Implementation_via_operator_discretization). In Wikipedia, The Free Encyclopedia.

- Fornberg, B. (1988). Generation of Finite Difference Formulas on Arbitrarily Spaced Grids. Mathematics of Computation, 51(184), 699-706. [https://doi.org/10.1090/S0025-5718-1988-0935077-0 ](https://doi.org/10.1090/S0025-5718-1988-0935077-0 )

Finite Difference Approximations of the Vector Potential:

- Feynman R., Leighton R., Sands M. (2011). [The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity](https://www.feynmanlectures.caltech.edu/III_21.html). In In The Feynman Lectures on Physics: The New Millennium Edition, Volume 3, chapter 21. Basic Books.

- Wikipedia contributors. (2021, April 21). [Peierls substitution](https://en.wikipedia.org/wiki/Peierls_substitution). In Wikipedia, The Free Encyclopedia.

Fast Fourier Transform:

- Press W. et al. (1992). Fast Fourier Transform.
In [Numerical Recipes in Fortran 77](https://websites.pmc.ucsc.edu/~fnimmo/eart290c_17/NumericalRecipesinF77.pdf), chapter 12.

- Wikipedia contributors. (2021, October 8). [Cooley–Tukey FFT algorithm](https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm). In Wikipedia, The Free Encyclopedia.

- Weisstein, E. (2021). [Fast Fourier Transform](https://mathworld.wolfram.com/FastFourierTransform.html). In Wolfram MathWorld.

Hartree Atomic Units:

- Wikipedia contributors. (2021, May 14). [Hartree atomic units](https://en.wikipedia.org/wiki/Hartree_atomic_units). In Wikipedia, The Free Encyclopedia.

Boundary Conditions and their Nomenclature:
 - Wikipedia contributors. (2021, March 7). [Boundary value problem](https://en.wikipedia.org/wiki/Boundary_value_problem). In Wikipedia, The Free Encyclopedia.

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