### Solution Of The Time Dependent Schrodinger Equation Using The Higher Order Trotter-Suzuki Method

Dhea Chita Octavianty

^{(1*)}, Pekik Nurwantoro

^{(2)}

(1) Universitas Gadjah Mada

(2) Universitas Gadjah Mada

(*) Corresponding Author

#### Abstract

We solve the solution of the Time-Dependent Schrodinger Equation (TDSE) using the higher-order Trotter-Suzuki Decomposition. We use The Gaussian function as a stationary state in a linear potential system. The TDSE solution using Baker - Campbell - Hausdorff was used to validate the results and to measure the accuracy of the Trotter - Suzuki decomposition. So that the difference between the Baker - Campbell - Hausdorff result and the Suzuki - Trotter decomposition is considered an error. The error of the TDSE solution by the Trotter - Suzuki second-order decomposition was lower than the first order. Meanwhile, the error of the TDSE solution by second-order hybrid will be lower than the second-order Trotter - Suzuki decomposition when the value of dx=0.1 and 0.05 with dt < 0.0001. The error comparison of these three methods is only valid when time t<1.

#### Keywords

#### Full Text:

PDF#### References

Ariasoca, T. A., Sholihun and Santoso, I., 2019, ‘Trotter-Suzuki-time propagation method for calculating the density of states of disordered graphene’, Computational Materials Science, 156, pp. 434–440.

Becerril, R., Guzman, F.S., Rendo-Romero, A. dan Valdez-Alvarado, S., 2008, ‘Solving the time-dependent Schrödinger equation using finite difference methods’, Revista Mexicana de Fisica E, 54, pp. 120–132.

Hatano, N. and Suzuki, M., 2005, ‘Finding Exponential Product Formulas of Higher Orders’, in Quantum Annealing and Other Optimization Methods, pp. 37–68.

De Raedt, H., 1987, ‘Product formula algorithms for solving the time dependent Schrödinger equation’, Computer Physics Reports, 7(1), pp. 1–72.

Soto-Eguibar, F. and Moya-Cessa, H. M., 2015, ‘Solution of the schr ödinger equation for a linear potential using the extended Baker-Campbell-Hausdorffformula’, Applied Mathematics and Information Sciences, 9, pp. 175–181.

Suzuki, M., 1976, ‘Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems’, Communications in Mathematical Physics, 51(2), pp. 183–190.

Suzuki, M., 1990, ‘Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations’, Physics Letters A, 146, pp. 319–323.

Suzuki, M., 1995, ‘Hybrid exponential product formulas for unbounded operators with possible applications to Monte Carlo simulations’, Physics Letters A.

Suzuki, M., 2000, ‘Mathematical basis of computational statistical physics and quantum analysis’, Computer Physics Communications, 127(1), pp. 32–36.

Suzuki, M. and Umeno, K., 1993, ‘Higher-Order Decomposition Theory of Exponential Operators and Its Applications to QMC and Nonlinear Dynamics’, in Computer Simulation Studies in Condensed-Matter Physics VI, pp. 74–86.

Wittek, P. and Cucchietti, F. M., 2013, ‘A second-order distributed Trotter-Suzuki solver with a hybrid kernel’, Computer Physics Communications, 184, pp. 1165–1171.

DOI: https://doi.org/10.22146/jfi.v26i1.66637

#### Article Metrics

Abstract views : 1178 | views : 959### Refbacks

- There are currently no refbacks.

Copyright (c) 2022 Dhea Chita Octavianty, Pekik Nurwantoro

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.