Quantum and semi-classical aspects of confined systems with variable mass (original) (raw)
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Journal of Physics A: Mathematical and Theoretical
The quantization of systems with a position dependent mass (PDM) is studied. We present a method that starts with the study of the existence of Killing vector fields for the PDM geodesic motion (Lagrangian with a PDM kinetic term but without any potential) and the construction of the associated Noether momenta. Then the method considers, as the appropriate Hilbert space, the space of functions that are square integrable with respect to a measure related with the PDM and, after that, it establishes the quantization, not of the canonical momenta p, but of the Noether momenta P instead. The quantum Hamiltonian, that depends on the Noether momenta, is obtained as an Hermitian operator defined on the PDM Hilbert space. In the second part several systems with position-dependent mass, most of them related with nonlinear oscillators, are quantized by making use of the method proposed in the first part.
Ambiguities in Quantizing a Classical System
One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.
Phase space quantum–classical hybrid model
Annals of Physics, 2019
In this work we provide a complete model of semiclassical theories by including back-reaction and correlation into the picture. We specially aim at the interaction between light and a two-level atom, and we also illustrate it via the coupling of two harmonic oscillators. Quantum and classical systems are treated on the same grounds via the Wigner-Weyl phase-space correspondence of the quantum theory. We show that this model provides a suitable mixture of the quantum and classical degrees of freedom, including the fact that the evolution transfers nonclassical features to the classical subsystem and nonquantum behavior to the quantum subsystem. In that sense, we can no longer distinguish between classical and quantum variables and we need to talk about a hybrid model.
Quantization of the 1-D harmonic oscillator with variable mass using the operators v̂ and p̂
2016
For the 1-D harmonic oscillator with position depending variable mass, a Hamiltonian and constant of motion are given through a consistent approach. Then, the quantization of this system is carried out using the operator p̂, for the Hamiltonian, and the operator v̂ for the constant of motion. We find that the result of both quantizations brings about different quantum dynamics.
On Quantization, the Generalized Schrödinger Equation and Classical Mechanics
UM-P-91/47, 1991
Using a new state-dependent, λ-deformable, linear functional operator, Q λ ψ , which presents a natural C ∞ deformation of quantization, we obtain a uniquely selected non-linear, integro-differential Generalized Schrödinger equation. The case Q 1 ψ reproduces linear quantum mechanics, whereas Q 0 ψ admits an exact dynamic, energetic and measurement theoretic reproduction of classical mechanics. All solutions to the resulting classical wave equation are given and we show that functionally chaotic dynamics exists.
Journal of Physics A: Mathematical and Theoretical
We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel-Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant k and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to ∞, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.
Semiclassical quantization of the Liouville formulation of classical mechanics
The Journal of Chemical Physics, 1988
A new method for the quantization of classical Hamiltonian systems is presented. This method is based upon the correspondence between the Liouville formulation of classical mechanics and the Liouville-von Neumann formulation of quantum mechanics. It does not distinguish between integrable and nonintegrable systems, and consequently, is equally applicable to both types of systems. Further, it treats the indistinguishability ofidentical particles correctly, and thus, the semiclassical eigenstates have the correct symmetry properties. Application of the method is illustrated by a series of examples. The results are in excellent agreement with quantum mechanics and represent an improvement over results obtained using the uniform semiclassical approximation.
Quantum Mechanics as a Classical Theory VIII: Second Quantization
2008
We continue in this paper our program of rederiving all quantum mechanical formalism from the classical one. We now turn our attention to the derivation of the second quantized equations, both for integral and half-integral spins. We then show that all the quantum results may be derived using our approach and also show the interpretation suggested by this derivation. This paper may be considered as a first approach to the study of the quantum field theory beginning by the same classical ideas we are supporting since the first paper of this series. 1