The phase of a quantum mechanical particle in curved spacetime

Abstract

We investigate the quantum mechanical wave equations for free particles of spin 0,1/2,1 in the background of an arbitrary static gravitational field in order to explicitly determine if the phase of the wavefunction is $S/\hslash =\int p_{\mu }d{x}^{\mu }/\hslash$, as is often quoted in the literature. We work in isotropic coordinates where the wave equations have a simple managable form and do not make a weak gravitational field approximation. We interpret these wave equations in terms of a quantum mechanical particle moving in medium with a spatially varying effective index of refraction. Due to the first order spatial derivative structure of the Dirac equation in curved spacetime, only the spin 1/2 particle has \textit{exactly} the quantum mechanical phase as indicated above. The second order spatial derivative structure of the spin 0 and spin 1 wave equations yield the above phase only to lowest order in $\hslash$. We develop a WKB approximation for the solution of the spin 0 and spin 1 wave equations and explore amplitude and phase corrections beyond the lowest order in $\hslash$. For the spin 1/2 particle we calculate the phase appropriate for neutrino flavor oscillations.

30 pages, no figures. Submitted to Gen.Rel.Grav 17 Oct 00

References in corpus (1)

• Neutrino Oscillations in Brans-Dicke Theory of Gravity

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