dorsal/arxiv
View SchemaTime-energy and time-entropy uncertainty relations in dissipative quantum dynamics
| Authors | Gian Paolo Beretta |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0511091 |
| URL | https://arxiv.org/abs/quant-ph/0511091 |
Abstract
We derive exact relations and general inequalities that extend the usual time-energy uncertainty relations from the domain of unitary Hamiltonian dynamics to that of dissipative dynamics as described by a broad class of linear and nonlinear evolution equations for the density operator. For non-dissipative dynamics, by using the Schroedinger inequality instead of the Heisenberg-Robertson inequality, we obtain a general exact time-energy uncertainty relation which is sharper than the usual Mandelstam-Tamm-Messiah relation $\tau_F\Delta_H\ge \hbar/2$. For simultaneous unitary/dissipative dynamics, the usual time-energy uncertainty relation is replaced by a less restrictive relation that depends on the characteristic time of dissipation, $\tau$, and the uncertainty associated with the generalized nonequilibrium Massieu-function operator which defines the structure of the dissipative part of the assumed class of evolution equations. Within the steepest-entropy-ascent dissipative quantum dynamics of an isolated system introduced earlier by this author, we obtain the interesting time-energy and time-entropy uncertainty relation $(2\tau_F\Delta_H/ \hbar)^2+ (\tau_F\Delta_S/k_B\tau)^2 \ge 1$. We illustrate this result and various other inequalities by means of numerical simulations.
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"abstract": "We derive exact relations and general inequalities that extend the usual\ntime-energy uncertainty relations from the domain of unitary Hamiltonian\ndynamics to that of dissipative dynamics as described by a broad class of\nlinear and nonlinear evolution equations for the density operator. For\nnon-dissipative dynamics, by using the Schroedinger inequality instead of the\nHeisenberg-Robertson inequality, we obtain a general exact time-energy\nuncertainty relation which is sharper than the usual Mandelstam-Tamm-Messiah\nrelation $\\tau_F\\Delta_H\\ge \\hbar/2$. For simultaneous unitary/dissipative\ndynamics, the usual time-energy uncertainty relation is replaced by a less\nrestrictive relation that depends on the characteristic time of dissipation,\n$\\tau$, and the uncertainty associated with the generalized nonequilibrium\nMassieu-function operator which defines the structure of the dissipative part\nof the assumed class of evolution equations. Within the steepest-entropy-ascent\ndissipative quantum dynamics of an isolated system introduced earlier by this\nauthor, we obtain the interesting time-energy and time-entropy uncertainty\nrelation $(2\\tau_F\\Delta_H/ \\hbar)^2+ (\\tau_F\\Delta_S/k_B\\tau)^2 \\ge 1$. We\nillustrate this result and various other inequalities by means of numerical\nsimulations.",
"arxiv_id": "quant-ph/0511091",
"authors": [
"Gian Paolo Beretta"
],
"categories": [
"quant-ph"
],
"title": "Time-energy and time-entropy uncertainty relations in dissipative quantum dynamics",
"url": "https://arxiv.org/abs/quant-ph/0511091"
},
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