The linear potential propagator via wave function expansion

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The Linear Potential Propagator via Wave Function Expansion

Antônio B. Nassar,^(1,2) José M. F. Bassalo,⁽²⁾ Paulo T. S. Alencar,⁽²⁾ José L. M. Lopes,⁽²⁾ José I. F. de Oliveira,⁽³⁾ and Mauro S. D. Cattani⁽⁴⁾

⁽¹⁾Extension Program-Department of Sciences, University of California, Los Angeles, California 90024, USA

⁽²⁾Departamento de Física da UFPA, 66075-900, Belém, Pará, Brazil

⁽³⁾Departamento de Física da UNAMA, 66050-000, Belém, Pará, Brazil

⁽⁴⁾Instituto de Física da USP, C.P. 66318, 05315-970, São Paulo, SP, Brazil

Received on 3 June, 2002

We evaluate the quantum propagator for the motion of a particle in a linear potential via a recently developped formalism [A.B. Nassar et al., Phys. Rev. E56, 1230, (1997)]. In this formalism, the propagator comes about as a type of expansion of the wave function over the space of the initial velocities.

I Introduction

The quantum propagator for the motion of a particle in a linear potential has been recently discussed by several authors[1-4] using different techniques. This arsenal of techniques has an immediate pedagogical appeal and can be valuable in some future calculation which may require the use of a nonstandard approach.

Here we present an alternative approach for the evaluation of the linear potential propagator via a new formalism developed by Nassar et al.[5] In this formalism the central result is

where

X(t) is the solution to the classical equation

and a(t) is the solution to the quantum equation

The terms on the right-hand side of Equations (4) and (5) are due the presence of the classical and the quantum potential, respectively.

Equation (1) provides the propagator as a type of expansion of the wave function over the space of the initial velocities. It is worth noticing the presence of the quantum potential in the quantum action (Equation (3)) and in the expansion coefficient (Equation (2)) through the function a(t).

II Linear Potential

In what follows, we calculate in some detail the quantum propagator for the linear potential V = - f x through the formalism given by Equations (1,2,3), therefore demonstrating its validity and usefulness as a new tool for future investigations.

First, we find the solutions to Equations (4) and (5), subject to the initial conditions

to be

where

Equations (7) and (8) allow us to carry out in full the arguments of the exponent of Equations (1,2,3) as follows. The first and the second arguments of Equation (2) can be expanded, respectively, as

Likewise, the quantum action (Equation (3)) can be expressed as

The propagator in Equation (1) is of the general form

where the coefficients A(t), B(t), C(t) and D(t) are, respectively:

The terms in (x - x_o)², (x + x_o), and independent of x and x_o are, respectively,

This allows us to write

which is the result found in Refs. [1-4].

We have presented a new approach for the evaluation of the linear potential propagator. In this formalism, the propagator comes about as a type of expansion of the wave function over the space of the initial velocities. This new approach has a pedagogical appeal and poses an alternative route for future research investigations.

• [1] B. R. Holstein, Am. J. Phys. 65, 414 (1997).
• [2] G. P. Arrighini, N. L. Durante and C. Guidotti, Am. J. Phys. 64, 1036 (1996).
• [3] R. W. Robinett, Am. J. Phys. 64, 803 (1996).
• [4] L. S. Brown and Y. Zhang, Am. J. Phys. 62, 806 (1994).
• [5] A. B. Nassar, J. M. F. Bassalo, P. T. S. Alencar, L. S. G. Cancela and M. Cattani, Phys. Rev. E56, 1230 (1997).

Publication Dates

• Publication in this collection
  04 Dec 2002
• Date of issue
  Sept 2002

History

• Received
  03 June 2002