Heisenberg Uncertainty Relation in Quantum Liouville Equation

We consider the quantum Liouville equation and give a characterization of the solutions which satisfy the Heisenberg uncertainty relation. We analyze three cases. Initially we consider a particular solution of the quantum Liouville equation: the Wigner transformf(x,v,t) of a generic solutionψ(x;t) of the Schrödinger equation. We give a representation ofψ(x,t) by the Hermite functions. We show that the values of the variances ofxandvcalculated by using the Wigner functionf(x,v,t) coincide, respectively, with the variances of position operatorX^and conjugate momentum operatorP^obtained using the wave functionψ(x,t). Then we consider the Fourier transform of the density matrixρ(z,y,t) =ψ∗(z,t)ψ(y,t). We find again that the variances ofxandvobtained by usingρ(z,y,t) are respectively equal to the variances ofX^andP^calculated inψ(x,t). Finally we introduce the matrix∥Ann′(t)∥and we show that a generic square-integrable functiong(x,v,t) can be written as Fourier transform of a density matrix, provided that the matrix∥Ann′(t)∥is diagonalizable

Classicality of a quantum oscillator

Gaussian quantum systems exhibit many explicitly quantum effects but can be simulated classically. Using both the Hilbert space (Koopman) and the phase-space (Moyal) formalisms we investigate how robust this classicality is. We find failures of consistency of the dynamics of a hybrid classical-quantum systems from both perspectives. By demanding that no unobservable operators couple to the quantum sector in the Koopmanian formalism, we show that the classical equations of motion act on their quantum counterparts without experiencing any back-reaction, resulting in non-conservation of energy in the quantum system. Using the phase-space formalism we study the short time evolution of the moment equations of a hybrid classical-Gaussian quantum system and observe violations of the Heisenberg Uncertainty Relation in the quantum sector for a broad range of initial conditions. We estimate the time scale for these violations, which is generically rather short. This inconsistency indicates that while many explicitly quantum effects can be represented classically, quantum aspects of the system cannot be fully masked. We comment on the implications of our results for quantum gravity

Exact number conserving phase-space dynamics of the M-site Bose-Hubbard model

The dynamics of M-site, N-particle Bose-Hubbard systems is described in quantum phase space constructed in terms of generalized SU(M) coherent states. These states have a special significance for these systems as they describe fully condensed states. Based on the differential algebra developed by Gilmore, we derive an explicit evolution equation for the (generalized) Husimi-(Q)- and Glauber-Sudarshan-(P)-distributions. Most remarkably, these evolution equations turn out to be second order differential equations where the second order terms scale as 1/N with the particle number. For large N the evolution reduces to a (classical) Liouvillian dynamics. The phase space approach thus provides a distinguished instrument to explore the mean-field many-particle crossover. In addition, the thermodynamic Bloch equation is analyzed using similar techniques.Comment: 11 pages, Revtex