Journal ArticleDOI
Quantum dynamics of a solvable nonlinear chiral model
M. Lakshmanan,K Eswaran +1 more
TLDR
In this article, a quantum mechanical analogue of a classical nonlinear system is shown to be exactly solvable and its energy levels and eigenfunctions are obtained completely, and the ordering problem that arises in the quantum mechanical case is overcome.Abstract:
The quantum mechanical analogue of a classical nonlinear system is shown to be exactly solvable and its energy levels and eigenfunctions are obtained completely. The symmetric version (k0=0) of this model is the SU(2)(X)SU(2) chiral invariant Lagrangian in the Gasiorowicz-Geffen coordinates. The radial part of the classical equation of motion (in both the symmetric and non-symmetric cases) admits simple harmonic bounded solutions and the bound state energies of the quantized system show a linear dependence on the coupling parameter lambda . It is shown that the Bohr-Sommerfeld quantization procedure reproduces the form of the correct bound state energy levels while a perturbation theoretic treatment gives the exact energy expressions. The ordering problem that arises in the quantum mechanical case is overcome.read more
Citations
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Journal ArticleDOI
Dynamical symmetries in a spherical geometry. I
TL;DR: In this paper, it is shown how to transform Poisson bracket algebras into Lie algebraic structures, those of the symmetry groups SO(N+1) and SU(N) respectively: the Hamiltonian of each system is expressed as a function of the Casimir operators of its symmetry group.
Journal ArticleDOI
A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators
TL;DR: In this paper, a super-integrable two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms, where all the bounded motions are quasiperiodic oscillations and the unbounded motions are represented by hyperbolic functions.
Journal ArticleDOI
A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour
TL;DR: In this article, the quantum version of a non-linear oscillator, previously analyzed at the classical level, is studied, and the λ-dependent Schrodinger equation is exactly solved as a Sturm-Liouville problem.
Journal ArticleDOI
Unusual Liénard-type nonlinear oscillator
TL;DR: A Liénard type nonlinear oscillator of the form x+kxx+(k2/9)x3+lambda1x=0, which may also be considered as a generalized Emden-type equation, is shown to possess unusual nonlinear dynamical properties.
Journal ArticleDOI
A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour
TL;DR: In this article, the quantum version of a non-linear oscillator was analyzed at the classical level and the Schr\"odinger equation was exactly solved as a Sturm-Liouville problem and the $\la$-dependent eigenenergies and eigenfunctions were obtained for both ≥ 0 and ≥ 0.
References
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Book
Practical quantum mechanics
TL;DR: In this paper, one-body problems without spin are discussed. And the Wentzel-Kramers Brillouin (WKB) approximation of the WKB is used.
Journal ArticleDOI
Effective lagrangians and field algebras with chiral symmetry.
S. Gasiorowicz,D.A. Geffen +1 more
TL;DR: In this paper, a review of effective Lagrangians and field algebras as means of treating chiral symmetry and partially conserved axial current (PCAC) for the study of elementary particle physics is presented.
Journal ArticleDOI
A quantum-mechanically solvable nonpolynomial Lagrangian with velocity-dependent interaction
P. M. Mathews,M. Lakshmanan +1 more
TL;DR: In this paper, the quantum-mechanical problem of the nonlinear oscillator with the Lagrangian L = 1/2 [x2-k0x2)/(l-λx2] is solved exactly and the energy levels and eigenfunctions are obtained completely.