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Bulletin of the Korean Chemical Society

An Analytical Solution of the Schrodinger Equation for a Rectangular Barrier with Time-Dependent Position

Tae Jun Park

Published Feb 20, 2002• 3 citations
Publisher

Abstract

Tae Jun ParkDepartment of Chemistry, Dongguk University, Seoul 100-715, KoreaReceived November 12, 2001Keywords : Kramers-Henneberger transformation, Time-dependent, Rectangular barrier.An analytical solution for the Schrodinger equation withtime-dependent potential has been investigated extensivelyover past decades. In addition to its own mathematicalinterest, this problem has wide applications in many areas ofphysics, such as laser-induced dynamics, the motion of Paultrap ions,

DOI
10.5012/bkcs.2002.23.2.357

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Tae Jun ParkDepartment of Chemistry, Dongguk University, Seoul 100-715, KoreaReceived November 12, 2001Keywords : Kramers-Henneberger transformation, Time-dependent, Rectangular barrier.An analytical solution for the Schrodinger equation withtime-dependent potential has been investigated extensivelyover past decades. In addition to its own mathematicalinterest, this problem has wide applications in many areas ofphysics, such as laser-induced dynamics, the motion of Paultrap ions,
Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.
We solve the Schrodinger equation analytically for systems whose potentials have a certain time-dependence (which is monotonously decreasing) and general coordinate-dependences.Only a few time-dependent systems have been reported to be analytically solved whose potentials are constant, linear, and quadratic functions of coordinate with arbitrary time-dependences.From a different perspective, we focus on the time-dependent systems whose potentials are monotonously decreasing functions of time with arbitrary coordinate dependences.Time-dependent potential of any coordinate-dependence can be handled analytically by transforming it to a time-independent potential of known solutions if its time-dependence is monotonously decreasing.We do this by a unitary transformation of the wavefunction and variable transformations to change the Schrodinger equation to be time-independent in new variables.These variables are then determined by solving a set of simple differential equations.This way we are able to find and to obtain analytical solutions for time-dependent potentials which we mention above.
Quantum mechanics describes phenomena that are key to understanding processes that cannot be explained by classical physics. This paper analyzes the behavior of particles in the presence of a potential barrier through numerical solutions to both time-dependent and time-independent Schrödinger equations. These results enabled a detailed understanding of the energy structure and temporal dynamics of wave functions, including the phenomenon of quantum tunneling. The obtained data have a wide range of applications in technology: from semiconductor devices to quantum computing.
Le soluzioni analitiche esatte dell'equazione di Schrodinger dipendente dal tempo sono interessanti perche possono aiutare ad esplorare ulteriormente vari affascinanti fenomeni quantistici o a chiarire alcuni concetti sottili. Ad oggi, le soluzioni trovate si riferiscono a pochi casi, quelli piu semplici e quelli in cui il potenziale assume una forma particolare. In questo lavoro si risolve analiticamente l'equazione di Schrodinger dipendente dal tempo considerando diverse tipologie di potenziali additivi dipendenti dal tempo mediante l'utilizzo di metodi di trasformazione, come ad esempio la trasformazione di tempo-spazio e quella di Kramers-Henneberger.