We discuss the Casimir effect for a massive bosonic field with mixed (Dirichlet-Neumann) boundary conditions. We use the ζ-function regularization prescription to obtain our physical results. Particularly, we analyse how the Casimir energy varies with the mass of the field and compare this mass dependence with those obtained for other boundary conditions. This is done graphically. Some other graphs involving a massive fermionic field are also included.

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We show that the difference of adiabatic phases, that are basis-dependent, in noncyclic evolution of nondegenerate quantum systems have to be taken into account to give the correct interference result in the calculation of physical quantities in states that are a superposition of instantaneous eigenstates of energy. To verify the contribution of those adiabatic phases in the interference phenomena, we consider the spin-1/2 model coupled to a precessing external magnetic field. In the model, the adiabatic phase increases in time up to reach the difference of the Berry's phases of the model when the external magnetic field completes a period.Keywords: Berry's phase, adiabatic phase, noncyclic adiabatic evolution, spin-1/2 model In 1928 Born and Fock[1] proofed the Adiabatic Theorem. In a quantum system with non-degenerate energy spectrum, this theorem says that if the system at t = 0 is an eigenstate of energy with quantum numbers {n}, along an adiabatic evolution it continues to be in an eigenstate of energy at time t with the same initial quantum numbers {n}. As a consequence of this theorem, the vector state of the quantum system acquires an extra phase besides the dynamical phase. This extra phase is actually named geometric phase. Before the important work by MV Berry in 1984[2] with cyclic adiabatic hamiltonian, this extra phase was realized to be dependent on the choice of the basis of instantaneous eigenstates of energy. This extra phase was considered non-physical since it could be absorbed in the choice of the states in the instantaneous basis. [3].In Ref.[2], MV Berry showed that the adiabatic phase acquired by the instantaneous eigenstates of energy, after a closed evolution in the classical parameter space, is physical due to its independence to the chosen basis to describe the state vector at each instant. Since the publication of the Ref.[2], the study of Berry's phase has followed very interesting and broad directions. More recently, the geometric phases have been proposed as a prototype for a quantum bit (qubit) [4][5][6][7]. In 1988 Samuel and Bhandari[8] generalized the geometric phase to noncyclic evolution. Many others interesting papers appear to discuss those physical phases in noncyclic evolution in the classical parameter space [9][10][11]. Experimental verification to the presence of those noncyclic geometric phases have been realized [12].The interference effect is a keystone in the linearity of the Quantum Mechanics. In the present letter we address to the question of the effect of the adiabatic evolution on the phases in quantum systems leaves a physical trace in measurable quantities associated to the noncyclic evolution of states described by a superposition of instantaneous eigenstates of energy. The same question was proposed in the nice Ref. [9], but differently from them we do not look for a physical noncyclic geometric phase. * Electronic address: mtt@if.uff.br Let us consider a time-dependent hamiltonian H(t) that evolves adiabatically.Following Ref.[2], we leave open the pos...

We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].Comment: 11 pages, no figur

We present explicitly another example of a temperature inversion symmetry in the Casimir effect for a nonsymmetric boundary condition. We also give an interpretation for our result. PACS numbers: 11.10.Wx, 12.20.Ds, This brief report was motivated by a recent paper published by Santos et al. [1], in which they discuss the temperature inversion symmetry in the Casimir effect [2] for mixed boundary conditions (for a detailed discussion on the Casimir effect see [3,4] and references therein). In an earlier paper, Ravndal and Tollefsen [5] showed that for the usual setup of two parallel plates a simple inversion symmetry arises in the Casimir effect at finite temperature. Temperature inversion symmetry also appeared in the Brown-Maclay work[6] where they related directly the zero-temperature Casimir energy to the energy density of blackbody radiation at temperature T . A few other papers on this kind of symmetry have also been published [7,8,9,10,11]. Until the publication of Ref. [1], this kind of inversion symmetry had appeared only in calculations of Casimir energy involving symmetric boundary conditions. In 1999, Santos et al. [1] showed, for the case of a massless scalar field submitted to mixed boundary conditions (Dirichlet-Neumann), that the Helmholtz free energy per unit area could be written as a sum of two terms, each of them obeying separately a temperature inversion symmetry. Our purpose here is to present another kind of nonsymmetric boundary condition for which there exists such a symmetry. We show * Electronic address: acap@if.ufrj.br † Electronic address: britto@if.ufrj.br ‡ Electronic address: fabiopr@if.ufrj.br § Electronic address: siqueira@if.ufrj.br

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We study the adiabatic limit in the density matrix approach for a quantum system coupled to a weakly dissipative medium. The energy spectrum of the quantum model is supposed to be non-degenerate. In the absence of dissipation, the geometric phases for periodic Hamiltonians obtained previously by M.V. Berry are recovered in the present approach. We determine the necessary condition satisfied by the coefficients of the linear expansion of the non-unitary part of the Liouvillian in order to the imaginary phases acquired by the elements of the density matrix, due to dissipative effects, be geometric. The results derived are model-independent. We apply them to spin 1 2 model coupled to reservoir at thermodynamic equilibrium.