The phase of a quantum mechanical particle in curved spacetime

Details The phase of a quantum mechanical particle in curved spacetime The phase of a quantum mechanical particle in curved spacetime

2000-10-18

arxiv.org/abs/gr-qc/0010065v1

Gen.Rel.Grav. 33 (2001) 1459-1487

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

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

Scientific paper

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 $S/\hbar = \int p_{\mu} dx^{\mu} / \hbar$, 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 $\hbar$. 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 $\hbar$. For the spin 1/2 particle we calculate the phase appropriate for neutrino flavor oscillations.

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