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Applied Mathematics & Information Sciences

Solution of the Schr�dinger Equation for a Linear Potential using the Extended Baker-Campbell-Hausdorff Formula

Francisco Soto, H. M. Moya-Cessa

Published Sep 8, 2014• 10 citations
Publisher

Abstract

The time-dependent Schr¨ odinger equation is solved for a linear potential using operational methods ; in particular, an extension of the Baker-Campbell-Hausdorff formula is exposed and used. Several initial conditions are considered. A closed form for the Wigner function is presented. The results can be extended to the propagation of an electromagnetic field in the paraxial approximation. odinger equation, linear potential, Zassenhaus formula, Baker-Campbell-Hausdorff formula, Wigner function

DOI
10.12785/amis/010123

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We consider the general Wigner function for a particle confined to a finite interval and subject to Dirichlet boundary conditions. We derive the boundary corrections to the “stargenvalue” equation and to the time evolution equation. These corrections can be cast in the form of a boundary potential contributing to the total Hamiltonian which together with a subsidiary boundary condition is responsible for the discretization of the energy levels. We show that a completely analogous formulation (in terms of boundary potentials) is also possible in standard operator quantum mechanics and that the Wigner and the operator formulations are also in one-to-one correspondence in the confined case. In particular, we extend Baker’s converse construction to bounded systems. Finally, we elaborate on the applications of the formalism to the subject of Wigner trajectories, namely in the context of collision processes and quantum systems displaying chaotic behavior in the classical limit.

arXiv:quant-ph/0012140•Journal of Mathematical Physics•Sep 26, 2002•43 citations
The time-dependent Schr¨ odinger equation is solved for a linear potential using operational methods ; in particular, an extension of the Baker-Campbell-Hausdorff formula is exposed and used. Several initial conditions are considered. A closed form for the Wigner function is presented. The results can be extended to the propagation of an electromagnetic field in the paraxial approximation. odinger equation, linear potential, Zassenhaus formula, Baker-Campbell-Hausdorff formula, Wigner function
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.
The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.
For the general time-dependent linear potential, the exact and complete solution of the Schr?dinger equation was obtained by assuming a form of wave function. Meanwhile, it was pointed out that the matrices φ(t)〉和ψ(t)〉 defined by the two wave functions 〈φ(t)xψ(t)〉和〈φ(t)pψ(t)〉 satisfy classical equations of motion. According to the ensemble theory of quantum mechanics, such classical equations describe the motion of fluid. A further research shows that similar conclusions apply to any linear system.
We consider the general Wigner function for a particle confined to a finite interval and subject to Dirichlet boundary conditions. We derive the boundary corrections to the “stargenvalue” equation and to the time evolution equation. These corrections can be cast in the form of a boundary potential contributing to the total Hamiltonian which together with a subsidiary boundary condition is responsible for the discretization of the energy levels. We show that a completely analogous formulation (in terms of boundary potentials) is also possible in standard operator quantum mechanics and that the Wigner and the operator formulations are also in one-to-one correspondence in the confined case. In particular, we extend Baker’s converse construction to bounded systems. Finally, we elaborate on the applications of the formalism to the subject of Wigner trajectories, namely in the context of collision processes and quantum systems displaying chaotic behavior in the classical limit.