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Revisiting Quantum Notions of Stress
I plan to submit the attached paper on quantum mechanical definition of stress in the next few weeks. Comments and feedback are welcome. Fair amount of work has been done on stress definition in the context of classical molecular dynamics (also attracting some controversies). In contrast, there appear to be several open issues in the quantum case. Hopefully, the attached paper provides a starting point.
Abstract: An important aspect of multiscale modeling of materials is to link continuum concepts such as fields to the underlying discrete microscopic behavior in a seamless manner. With the growing importance of atomistic calculations to understand material behavior, reconciling continuum and discrete concepts is necessary to interpret molecular and quantum mechanical simulations. In this work, we provide a quantum mechanical framework to a distinctly continuum quantity: mechanical stress. While the concept of the global macroscopic stress tensor in quantum mechanics has been well established, there still exist open issues when it comes to a spatially varying local quantum stress tensor. We attempt to shed some light on this topic by establishing a general quantum mechanical operator based approach to continuity equations and from those, introduce a local quantum mechanical stress tensor. Further, we elucidate the analogies that exist between (classical) molecular dynamics based stress definition and the quantum stress. Our derivations seem to suggest that the local quantum mechanical stress may not be an observable in quantum mechanics and therefore traces the non-uniqueness of the atomistic stress tensor to the gauge arbitrariness of the quantum mechanical state-function. Lastly, the virial stress theorem (of empirical molecular dynamics) is re-derived in a transparent manner that elucidates the analogy between quantum mechanical global stress.