Energy densities in quantum mechanics
- URL: http://arxiv.org/abs/2305.05657v3
- Date: Thu, 4 Jan 2024 13:56:38 GMT
- Title: Energy densities in quantum mechanics
- Authors: V. Stepanyan and A.E. Allahverdyan
- Abstract summary: We start from a possibly fundamental, relativistic description for a spin-$frac12$ particle: Dirac's equation.
We find a locally conserved non-relativistic energy density that is defined via the Terletsky-Margenau-Hill quasiprobability.
We find a new form of spin-related energy that is finite in the non-relativistic limit, emerges from the rest energy, and is (separately) locally conserved.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Quantum mechanics does not provide any ready recipe for defining energy
density in space, since the energy and coordinate do not commute. To find a
well-motivated energy density, we start from a possibly fundamental,
relativistic description for a spin-$\frac{1}{2}$ particle: Dirac's equation.
Employing its energy-momentum tensor and going to the non-relativistic limit we
find a locally conserved non-relativistic energy density that is defined via
the Terletsky-Margenau-Hill quasiprobability (which is hence selected among
other options). It coincides with the weak value of energy, and also with the
hydrodynamic energy in the Madelung representation of quantum dynamics, which
includes the quantum potential. Moreover, we find a new form of spin-related
energy that is finite in the non-relativistic limit, emerges from the rest
energy, and is (separately) locally conserved, though it does not contribute to
the global energy budget. This form of energy has a holographic character,
i.e., its value for a given volume is expressed via the surface of this volume.
Our results apply to situations where local energy representation is essential;
e.g. we show that the energy transfer velocity for a large class of free
wave-packets (including Gaussian and Airy wave-packets) is larger than its
group (i.e. coordinate-transfer) velocity.
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