Abstract
The Aharonov-Vaidman gauge additively transforms the mean energy of a quantum mechanical system into a weak valued system energy. In this paper, the equation of motion of this weak valued energy is used to provide a mathematical statement of an extended 1st Law of Thermodynamics that is applicable to the mean energy of a closed quantum system when the mean energy is expressed in the Aharonov-Vaidman gauge, i.e., when the system’s energy is weak valued. This is achieved by identifying the generalized heat and work exchange terms that appear in the equation of motion for weak valued energy. The complex valued contributions of the additive gauge term to these generalized exchange terms are discussed and this extended 1st Law is shown to subsume the usual 1st Law that is applicable for the mean energy of a closed quantum system. It is found that the gauge transformation introduces an additional energy uncertainty exchange term that—while it is neither a heat nor a work exchange term—is necessary for the conservation of weak valued energy. A spin-1/2 particle in a uniform magnetic field is used to illustrate aspects of the theory. It is demonstrated for this case that the extended 1st Law implies the existence of a gauge potential ω and that it generates a non-vanishing gauge field F. It is also shown for this case that the energy uncertainty exchange accumulated during the evolution of the system along a closed evolutionary cycle C in an associated parameter space is a geometric phase. This phase is equal to both the path integral of ω along C and the integral of the flux of F through the area enclosed by C.
Highlights
State pre-selection and post-selection (PPS) techniques have been used in recent years to manipulate and control quantum systems in such diverse research areas as quantum system-environment interactions (e.g., [1]), the quantum eraser (e.g., [2]), and Pancharatnam phase (e.g., [3,4])
Where and are continuous functions forming the top and bottom portions of the boundary of. Using this in Corollary 6.5 yields sin 2 sec sin 2 sec sin 2 sec or sin 2 sec Applying this approach to Theorem 6.1 yields the same result, i.e., sin 2 sec sin 2 sec Closed weak valued energy quantum systems have been shown to conform to a 1st Law of Thermodynamics that is an extension of the usual 1st Law of Thermodynamics for closed mean energy quantum systems
Energy uncertainty exchange term that is required for weak valued energy conservation appears in the mathematical statement of the extended 1st Law
Summary
State pre-selection and post-selection (PPS) techniques have been used in recent years to manipulate and control quantum systems in such diverse research areas as quantum system-environment interactions (e.g., [1]), the quantum eraser (e.g., [2]), and Pancharatnam phase (e.g., [3,4]). A PPS defined uncertainty quantity called the Aharonov-Vaidman (AV) gauge was introduced as a new “scale of measurement” for the mean values of quantum mechanical observables [29] This gauge additively transforms a mean value of an observable into an associated weak value and induces each of the eccentric characteristics exhibited by weak values. The extended 1st Law is illustrated via its application to a spin-1/2 particle in a uniform magnetic field For this system it is shown that: (i) the extended 1st Law implies the existence of a gauge potential and an associated non-vanishing gauge field; and (ii) the energy uncertainty exchange accumulated during a cyclic evolution of the system in a requisite parameter space is a geometric phase.
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