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.
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...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.