Energy densities in quantum mechanics

Quantum mechanics does not provide any ready recipe for defining energydensity in space, since the energy and coordinate do not commute. To find awell-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 wefind a locally conserved non-relativistic energy density that is defined viathe Terletsky-Margenau-Hill quasiprobability (which is hence selected amongother options). It coincides with the weak value of energy, and also with thehydrodynamic energy in the Madelung representation of quantum dynamics, whichincludes the quantum potential. Moreover, we find a new form of spin-relatedenergy that is finite in the non-relativistic limit, emerges from the restenergy, and is (separately) locally conserved, though it does not contribute tothe 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 freewave-packets (including Gaussian and Airy wave-packets) is larger than itsgroup (i.e. coordinate-transfer) velocity.