Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLorentz Invariant Berry Phase for a Perturbed Relativistic Four Dimensional Harmonic Oscillator
145
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We show the existence of Lorentz invariant Berry phases generated, in the Stueckleberg-Horwitz-Piron manifestly covariant quantum theory (SHP), by a perturbed four dimensional harmonic oscillator. These phases are associated with a fractional perturbation of the azimuthal symmetry of the oscillator. They are computed numerically by using time independent perturbation theory and the definition of the Berry phase generalized to the framework of SHP relativistic quantum theory.
rate research
Read More
The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators, one of which is the Hamiltonian. We show that the joint spectrum of the Hamilton operator, the $z$ component of the angular momentum, and a quartic integral obtained from separation in prolate spheroidal coordinates has quantum monodromy for sufficiently large energies. This means that one cannot globally assign quantum numbers to the joint spectrum. The effect can be classically explained by showing that the corresponding Liouville integrable system has a non-degenerate focus-focus point, and hence Hamiltonian monodromy.
As a generalization and extension of our previous paper {it J. Phys. A: Math. Theor. 53 055302} cite{AME2020}, in this work we study a quantum 4-body system in $mathbb{R}^d$ ($dgeq 3$) with quadratic and sextic pairwise potentials in the {it relative distances}, $r_{ij} equiv {|{bf r}_i - {bf r}_j |}$, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum ($S$-states). In variables $rho_{ij} equiv r_{ij}^2$, the corresponding reduced Hamiltonian of the system possesses a hidden $sl(7;{bf R})$ Lie algebra structure. In the $rho$-representation it is shown that the 4-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly-solvable (ES). We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite, three others are equal), molecular two-center (two masses are infinite, two others are equal) and molecular three-center (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born-Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. Also, the reduction to the lower dimensional cases $d=1,2$ is discussed. It is shown that for four body harmonic oscillator case there exists an infinite family of eigenfunctions which depend on the single variable which is the moment-of-inertia of the system.
The measurement of a quantum system becomes itself a quantum-mechanical process once the apparatus is internalized. That shift of perspective may result in different physical predictions for a variety of reasons. We present a model describing both system and apparatus and consisting of a harmonic oscillator coupled to a field. The equation of motion is a quantum stochastic differential equation. By solving it we establish the conditions ensuring that the two perspectives are compatible, in that the apparatus indeed measures the observable it is ideally supposed to.
We decorate the one-dimensional conic oscillator $frac{1}{2} left[-frac{d^{2} }{dx^{2} } + left|x right| right]$ with a point impurity of either $delta$-type, or local $delta$-type or even nonlocal $delta$-type. All the three cases are exactly solvable models, which are explicitly solved and analysed, as a first step towards higher dimensional models of physical relevance. We analyse the behaviour of the change in the energy levels when an interaction of the type $-lambda,delta(x)$ or $-lambda,delta(x-x_0)$ is switched on. In the first case, even energy levels (pertaining to antisymmetric bound states) remain invariant with $lambda$ although odd energy levels (pertaining to symmetric bound states) decrease as $lambda$ increases. In the second, all energy levels decrease when the form factor $lambda$ increases. A similar study has been performed for the so called nonlocal $delta$ interaction, requiring a coupling constant renormalization, which implies the replacement of the form factor $lambda$ by a renormalized form factor $beta$. In terms of $beta$, even energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of $beta$ the energy of each odd level, with the natural exception of the first one, becomes lower than the constant energy of the previous even level. Finally, we consider an interaction of the type $-adelta(x)+bdelta(x)$, and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.
We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyls one, does not.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
