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Chinese Physics C

Nonlinear Spectrum and Generating Algebra for Infinitely Deep Square Well Potential and New Coherent State

Ni Zhi

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

Using the Hamiltonian of symmetrical one dimersional infinitely deep square well potential and natural operators,we obtain its nonlinear spectrum and generating algebra,and get a class of new nonlinear coherent states on the basis of the nonlinear algebra obfained. These coherent states are of temporal stability,and can be regarded as the eigenstates of the lower operator with the eigenvalues in an operator field and as the rotational states of the generalized minimal uncertainly states as well.

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Using the Hamiltonian of symmetrical one dimersional infinitely deep square well potential and natural operators,we obtain its nonlinear spectrum and generating algebra,and get a class of new nonlinear coherent states on the basis of the nonlinear algebra obfained. These coherent states are of temporal stability,and can be regarded as the eigenstates of the lower operator with the eigenvalues in an operator field and as the rotational states of the generalized minimal uncertainly states as well.
In this work we make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) As eigenstates of a deformed annihilation operator and ii) by application of a deformed displacement operator to the vacuum state. We also construct the coherent states for the same systems using the ladder operators obtained by traditional methods with the knowledge of the eigenfunctions and eigenvalues of the corresponding Schrödinger equation. We show that both methods yield coherent states with identical algebraic structure.
Using the analytic representation of the so-called Gazeau–Klauder coherent states (CSs), we shall demonstrate that how a new class of generalized CSs, namely the family of dual states associated with theses states, can be constructed through viewing these states as temporally stable nonlinear CSs. Also we find that the ladder operators, as well as the displacement type operator corresponding to these two pairs of generalized CSs, may be easily obtained using our formalism, without employing the supersymmetric quantum mechanics (SUSYQM) techniques. Then, we have applied this method to some physical systems with known spectrum, such as Pöschl–Teller, infinite well, Morse potential and hydrogenlike spectrum as some quantum mechanical systems. Finally, we propose the generalized form of the Gazeau–Klauder CS and the corresponding dual family.
In this work, we construct nonlinear coherent states for Hamiltonians with linear and quadratic terms in the number operator by the generalization of two definitions: as eigenstates of a deformed annihilation operator and as those states obtained by the application of a deformed displacement operator on the vacuum state. We evaluate their temporal dependence, analyze its dispersion relations and some of their statistical properties for two model Hamiltonians, one supporting a finite number of bound states and the other supporting an infinite number of bound states.