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Register a new userDisplacement operator for quantum systems with position-dependent mass
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Added by
Raimundo Costa Filho
Publication date
2011
fields
Physics
and research's language is
English
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A translation operator is introduced to describe the quantum dynamics of a position-dependent mass particle in a null or constant potential. From this operator, we obtain a generalized form of the momentum operator as well as a unique commutation relation for $hat x$ and $hat p_gamma$. Such a formalism naturally leads to a Schrodinger-like equation that is reminiscent of wave equations typically used to model electrons with position-dependent (effective) masses propagating through abrupt interfaces in semiconductor heterostructures. The distinctive features of our approach is demonstrated through analytical solutions calculated for particles under null and constant potentials like infinite wells in one and two dimensions and potential barriers.
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We study the $(1+1)$ dimensional generalized Dirac oscillator with a position-dependent mass. In particular, bound states with zero energy as well as non zero energy have been obtained for suitable choices of the mass function/oscillator interaction. It has also been shown that in the presence of an electric field, bound states exist if the magnitude of the electric field does not exceed a critical value.
An inhomogeneous Kaluza-Klein compactification to four dimensions, followed by a conformal transformation, results in a system with position dependent mass (PDM). This origin of a PDM is quite different from the condensed matter one. A substantial generalization of a previously studied nonlinear oscillator with variable mass is obtained, wherein the position dependence of the mass of a nonrelativistic particle is due to a dilatonic coupling function emerging from the extra dimension. Previously obtained solutions for such systems can be extended and reinterpreted as nonrelativistic particles interacting with dilaton fields, which, themselves, can have interesting structures. An application is presented for the nonlinear oscillator, where within the new scenario the particle is coupled to a dilatonic string.
The classical Einstein-Hilbert (EH) action for general relativity (GR) is shown to be formally analogous to the classical system with position-dependent mass (PDM) models. The analogy is developed and used to build the covariant classical Hamiltonian as well as defining an alternative phase portrait for GR. The set of associated Hamiltons equations in the phase space is presented as a first-order system dual to the Einstein field equations. Following the principles of quantum mechanics, I build a canonical theory for the classical general. A fully consistent quantum Hamiltonian for GR is constructed based on adopting a high dimensional phase space. It is observed that the functional wave equation is timeless. As a direct application, I present an alternative wave equation for quantum cosmology. In comparison to the standard Arnowitt-Deser-Misner(ADM) decomposition and quantum gravity proposals, I extended my analysis beyond the covariant regime when the metric is decomposed into the $3+1$ dimensional ADM decomposition. I showed that an equal dimensional phase space can be obtained if one applies ADM decomposed metric.
We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator, associated with a position-dependent effective mass, by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass, uniform field interactions and to obtain a generalized uncertainty relation (GUP) that behaves as a distinguishability measure for the coherent states, these latter satisfying a periodic evolution of the GUP corrections.
A recursion technique of obtaining the asymptotical expansions for the bound-state energy eigenvalues of the radial Schrodinger equation with a position-dependent mass is presented. As an example of the application we calculate the energy eigenvalues for the Coulomb potential in the presence of position-dependent mass and we derive the inequalities regulating the shifts of the energy levels from their constant-mass positions.
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