# IJPAM: Volume 49, No. 2 (2008)

**QUANTUM MECHANICS AND**

THE WEAK EQUIVALENCE PRINCIPLE

THE WEAK EQUIVALENCE PRINCIPLE

Departamento de Matemáticas

Universidad Nacional de Colombia

Bogotá, COLOMBIA

e-mail: rshuerfanob@unal.edu.co

Instituto de Ciencias Nucleares

Universidad Nacional Autónoma de México

Circuito Exterior, Ciudad Universitaria

México D.F., 04510, MÉXICO

e-mails: sarira@nucleares.unam.mx

e-mail: socolovs@nucleares.unam.mx

**Abstract.**We use the Feynman path integral approach to nonrelativistic quantum
mechanics twofold. First, we derive the Lagrangian for a spinless particle
moving in a uniformly but not necessarily constantly accelerated reference
frame; then, applying the *strong equivalence principle* (SEP) we obtain
the Schrödinger equation for a particle in an inertial
frame and in the presence of a uniform and constant gravity field. Second,
using the associated Feynman propagator, we propagate an initial Gaussian wave
packet, with the final wave function and probability density depending on the
ratio
, where is the inertial mass of the particle,
thus exhibiting the fact that *the weak equivalence principle* (WEP) *is violated by quantum mechanics*. The probability density is
well defined and mass independent in the classical limit, showing that in this limit
the WEP is recovered. At the quantum level a heavier particle does not
necessarily falls faster than a lighter one, depending on the
relations between the initial and final positions of
the particles. Due to the Ehrenfest's Theorem, however, the
expectation values of the physical observables obey classical
equations of motion; so, in the average, heavier
and lighter particles fall at the same rate.

**Received: **August 16, 2007

**AMS Subject Classification: **81Sxx

**Key Words and Phrases: **path integrals, weak and strong equivalence principles, nonrelativistic quantum mechanics, classical limit

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2008

**Volume:** 49

**Issue:** 2