Impenetrable barriers in quantum mechanics

S. De Vincenzo

Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47145, Caracas 1041–A, Venezuela, e–mail: svincenz@fisica.ciens.ucv.ve

Recibido el 24 de noviembre de 2006
Aceptado el 18 de septiembre de 2007

Abstract

We derive the expression V(x) u(x) = cδ (x — a) + v(x) u(x) (where V(x) is the potential, u(x) the wave function, c a constant and v(x) a finite potential function for x < a), which is present in the one–dimensional Schrödinger equation on the whole real line when we have an impenetrable barrier at x > a, that is, an infinite step potential there. By studying the solution of this equation, we identify, connect and discuss three different Hamiltonian operators that describe the barrier. We extend these results by constructing an infinite square–well potential from two impenetrable barriers.

Keywords: Quantum mechanics; Schrödinger equation; impenetrable barriers.

Resumen

Derivamos la expresión V(x) u(x) = cδ (x – a) + v(x) u(x) (donde V(x) es el potencial, u(x) la funcion de onda, c una constante y v(x) una función potencial finita para x < a), la cual se presenta en la ecuación de Schrödinger unidimensional sobre toda la línea real cuando se tiene una barrera impenetrable en x > a, es decir, un potencial salto infinito allí. Estudiando la solución de esta ecuación, identificamos, conectamos y discutimos tres diferentes operadores hamiltonianos que describen la barrera. Extendemos estos resultados al construir un potencial de pozo cuadrado infinito a partir de dos barreras impenetrables.

Descriptores: Mecánica cuántica; ecuación de Schrödinger; barreras impenetrables.

PACS: 03.65.–w

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Acknowledgements

I would like to thank the last anonymous referee for important comments and suggestions which led to improvements in the manuscript. Likewise, I would like to thank my relatives, as well my wife's relatives, in Italy, for their hospitality during summer 2007. In the time dedicated to this work, financial support was received from CDCH–UCV (project PI 03–00–6038–2005).

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