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Journal of Mathematical Chemistry

Study of uncertainty, average energy, and thermodynamic quantities of the time dependent quantum system

Debraj Nath

Published Aug 13, 2022• 7 citations
Publisher
DOI
10.1007/s10910-022-01388-x

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Exact solutions of time-dependent Schrödinger equation in presence of time-dependent potential is defined by point transformation and separation of variables. Energy and Heisenberg uncertainty relation are pursued for time-independent potential whereas average energy and Heisenberg uncertainty relation are defined for time-dependent potential. Forces acting on a fixed boundary wall as well as average force acting on moving boundary wall are presented along various trajectories. For high temperature, analytical forms of partition function and the corresponding thermodynamic quantities are derived following the Euler-Maclaurin summation formula over a finite as well as an infinite domain for accurate presentation. Three quantum systems are generated with the help of point transformation, separation of variables and super-symmetric quantum mechanics from one quantum system and the corresponding results are compared among all systems, where two of them are time-independent and another two are time-dependent.
The Heisenberg uncertainty relation, ΔxΔpx ⩾ ℏ/2 (ℏ ≡ h/2π), is well known. The purpose of this paper is to study a linkage between macroscopic thermodynamic quantities and the uncertainty relation. To simplify matters, we consider one-dimensional harmonic oscillator model. The calculations are carried out by two steps. First, the thermal average of the uncertainty relation is obtained for canonical system in thermal equilibrium at temperature T, which is denoted by (Δx)T(Δpx)T. Second, the thermodynamic quantities are expressed analytically as a function of (Δx)T(Δpx)T within the harmonic oscillator model. Finally, a connection between the energy fluctuations and the uncertainty relation is clarified for the Einstein’s heat capacity model. The analysis is made on the basis of fluctuation theory. The uncertainty relation is valid not only in quantum state but also at finite temperature.
Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems.