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arXiv:cond-mat/0407092 · Journal of Physics B Atomic Molecular and Optical Physics

On the mechanical properties of matter waves

Nicholas K Whitlock, Stephen M Barnett, John Jeffers

Published Jul 2, 2004• 3 citations
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Abstract

Electromagnetic waves and fluids have locally conserved mechanical properties associated with them and we may expect these to exist for matter waves. We present a semiclassical description of the continuity equations relating to these conserved properties of matter waves and derive a general expression for their respective fluxes.
DOI
10.1088/0953-4075/37/14/l01

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Maxwell matter waves emerge from a perspective, complementary to de Broglie's, that matter is fundamentally a wave phenomenon whose particle aspects are revealed by quantum mechanics. Their quantum mechanical description is derived through the introduction of a matter vector potential, having frequency ω0ω_0ω0​, to Schrodinger's equation for a massive particle. Maxwell matter waves are then seen to be coherent excitations of a single-mode of the matter-wave field. In the classical regime, their mechanics is captured by a matter analog of Maxwell's equations for the electromagnetic field. As such, Maxwell matter waves enable a spectrum of systems that have useful optical analogs, such as resonant matter-wave interferometric sensors and matter-wave parametric oscillators. These waves are associated with a wavelength that is tied to the drive frequency ω0ω_0ω0​ rather than to the massive particle's energy, as is ordinarily the case with de Broglie matter waves. As a result, simple interferometric measurements lead to different outcomes for the two types of waves. While their apparent departure from de Broglie character is surprising, Maxwell matter waves are wholly consistent with quantum mechanics.
PACS number: 03.75.-b Abstract. Electromagnetic waves and fluids have locally conserved mechanical properties associated with them and we may expect these to exist for matter waves. We present a semiclassical description of the continuity equations relating to these conserved properties of matter-waves and derive a general expression for their respective fluxes. It has long been known that electromagnetic waves have associated mechanical properties. Maxwell described the radiation pressure exerted by light [1] and Poynting identified the momentum carried by light [2] and that angular momentum was associated with circular polarisation [3]. The energy, momentum and angular momentum of electromagnetic waves are particularly important as these quantities are all conserved. Associated with the density of each of these quantities is a flux density (or current density) which is related to the density via continuity equations [4]. The creation of the first Bose-Einstein condensates (BECs) in dilute atomic gases in 1995 [5, 6, 7] brought about the possibility of creating coherent matter waves. A BEC in a trap is analogous to an electromagnetic field in a laser cavity, in that macroscopic occupation of a single quantum state occurs. Just as a laser cavity is a source of coherent light, a BEC can be a source of coherent matter waves. To continue this analogy we assert that matter waves will have associated mechanical properties and present a general theory of the local mechanical properties of matter waves. We give, as examples, the densities and flux densities of the matter wave particle number and energy. The same theory applies to their momentum and angular momentum. The local conservation of particles is a well established concept in fluid mechanics. It states that in any volume element of a fluid, no matter how small, the rate of change of particles in an element of fluid is equal to the rate at which the particles flow into the surface of the element. This principle is made quantitative in the following equation [8];
Abstract A general, multi-component Eulerian fluid theory is a set of nonlinear, hyperbolic partial differential equations. However, if the fluid is to be the large-scale description of a short-range many-body system, further constraints arise on the structure of these equations. Here we derive one such constraint, pertaining to the free energy fluxes. The free energy fluxes generate expectation values of currents, akin to the specific free energy generating conserved densities. They fix the equations of state and the Euler-scale hydrodynamics, and are simply related to the entropy currents. Using the Kubo–Martin–Schwinger relations associated to many conserved quantities, in quantum and classical systems, we show that the associated free energy fluxes are perpendicular to the vector of inverse temperatures characterising the state. This implies that all entropy currents can be expressed as averages of local observables. In few-component fluids, it implies that the averages of currents follow from the specific free energy alone, without the use of Galilean or relativistic invariance. In integrable models, in implies that the thermodynamic Bethe ansatz must satisfy a unitarity condition. The relation also guarantees physical consistency of the Euler hydrodynamics in spatially-inhomogeneous, macroscopic external fields, as it implies conservation of entropy, and the local-density approximated Gibbs form of stationarity states. The main result on free energy fluxes is based on general properties such as clustering, and we show that it is mathematically rigorous in quantum spin chains.