A quantum-mechanically solvable nonpolynomial Lagrangian with velocity-dependent interaction

Mathews, P. M. ; Lakshmanan, M. (1975) A quantum-mechanically solvable nonpolynomial Lagrangian with velocity-dependent interaction IL Nuovo Cimento A (1971-1996), 26 (3). pp. 299-316. ISSN 0369-3546

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Official URL: http://www.springerlink.com/content/65587315878273...

Related URL: http://dx.doi.org/10.1007/BF02769015


The quantum-mechanical problem of the nonlinear oscillator with the Lagrangian L= ½[x.2-k0x 2)/(l-λx2)] is solved exactly and the energy levels and eigenfunctions are obtained completely. This model (whenk 0=0) is the zero-space-dimensional isoscalar analogue of the nonlinear SU2⊗SU2 chirally invariant Lagrangian in the Gasiorowicz-Geffen co-ordinates and may also be considered as a modified version of the anharmonic-oscillator and Lee-Zumino models. The bound-state energy levels are found to have a linear dependence on the coupling parameter, in sharp contrast to the case of the familiar oscillator With quartic anharmonicity where the energy, as a function of λ, has complicated singularities at λ = 0. We investigate how far certain standard approximation procedures reproduce the exact results. The Bohr-Sommerfeld quantization procedure is found to reproduce the form of the boundstate energy levels correctly. Interestingly a perturbation-theoretic treatment also reproduces the correct results at least up to the order (λ2) to which we have carried our calculations.

Item Type:Article
Source:Copyright of this article belongs to Italian Physical Society.
ID Code:19582
Deposited On:22 Nov 2010 12:21
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