Energy densities in quantum mechanics

V. Stepanyan1 and A.E. Allahverdyan1,2

1Institute of Physics, Yerevan State University, 0025 Yerevan, ArmeniaAlikhanian National Laboratory, 0036 Yerevan, Armenia
2Energy densities in quantum mechanics

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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.

The definition of space-dependent energy density in quantum mechanics is not unique, because energy and coordinates do not commute and cannot be measured simultaneously. Nevertheless, defining energy density in a possibly clear way is and has been crucial in developing a new window into non-equilibrium quantum physics. As a starting point for defining this energy density, we take the relativistic Dirac's equation, which is possibly the fundamental description for a particle with spin one-half. By utilizing the energy-momentum tensor from Dirac's equation and taking the non-relativistic limit, we derive a locally conserved non-relativistic energy density. An important feature of this density is that its kinetic part should be locally negative for normalized wave packets (though its total value is positive). For several most common physical wave packets (e.g. Gaussian, Airy) this energy density has a higher transfer velocity than the coordinate velocity (i.e. group velocity) of the same wavepacket.

When deriving this energy density from Dirac's equation, we identify a new form of spin-related energy density, which is finite in the non-relativistic limit and emerges from the rest energy. This energy is locally conserved but it nullifies for most simple quantum mechanical states. Moreover, its total value is always zero so it has no contribution to the global energy of the particle. It is a holographic property, meaning that its volumetric value depends on its surface. This new energy density is thus worth studying and identifying in experiments.

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[3] Francisco Ricardo Torres Arvizu, Adrian Ortega, and Hernán Larralde, "On the energy density in quantum mechanics", Physica Scripta 98 12, 125015 (2023).

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