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Journal of Physical Studies 19(1/2), Article 1003 [10 pages] (2015)
DOI: https://doi.org/10.30970/jps.19.1003 OPERATORS AND BILINEAR DENSITIES IN THE DIRAC FORMAL 1D EHRENFEST THEOREMSalvatore De Vincenzo
Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, |
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We calculate formal time derivatives of the mean values of the standard position, velocity, and mechanical momentum operators, i. e., the Ehrenfest theorem for a one-dimensional Dirac particle in the coordinate representation. We show that these derivatives contain boundary terms that essentially depend on the values taken there by characteristic bilinear densities. We do not automatically take the boundary terms to vanish (as is usually done); nevertheless, we relate the boundary terms to similar terms that must be zero if one requires the hermiticity of certain specific unbounded operators. Throughout the article, we thoroughly discuss and illustrate all these aspects, which include the relations to certain boundary conditions. To clarify, we call our approach formal because all operations involving operators (for example, some operators products) are performed without respecting the restrictions imposed by the sets of functions on which the self-adjoint operators can act. Moreover, the Dirac Hamiltonian that we consider in our calculations contains a potential that is the time component of a Lorentz two-vector; nevertheless, we also obtain and concisely discuss the Ehrenfest theorem for a Hamiltonian with the most general Lorentz potential in (1+1) dimensions.
PACS number(s): 03.65.-w, 03.65.Ca, 03.65.Pm