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Journal of Jinggangshan University

CLASSIC REGULARIZATION OF NONLINEAR SCHRODINGER EQUATION AND CANONICAL TRANSFORMATION

Minglun Chen

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

It is well known that quantum system can be translated to classic Hamilton system strictly. We have proven that phase variables of classic Hamiltonian can be described as the form of phase difference variables because quantum system is of gauge transformation. Using these independent variables of (pa,Qa) (a=1,… ,n-1) where pa is the probability amplitude of ath component of wave function and Qa is the phase difference between ath and nth component. Schrodinger equations can be described as Hamilton equations. Furthermore,it has been determined that the variables of probability amplitude and phase difference between ath and nth component are not canonical variables. For two dimension quantum system,as the transformation of phase difference unchanged,it can be tested that the couples variable ((pa-pn)/2,Qa) (a=1,n=2) are another canonical variable.

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It is well known that quantum system can be translated to classic Hamilton system strictly. We have proven that phase variables of classic Hamiltonian can be described as the form of phase difference variables because quantum system is of gauge transformation. Using these independent variables of (pa,Qa) (a=1,… ,n-1) where pa is the probability amplitude of ath component of wave function and Qa is the phase difference between ath and nth component. Schrodinger equations can be described as Hamilton equations. Furthermore,it has been determined that the variables of probability amplitude and phase difference between ath and nth component are not canonical variables. For two dimension quantum system,as the transformation of phase difference unchanged,it can be tested that the couples variable ((pa-pn)/2,Qa) (a=1,n=2) are another canonical variable.
General Hamiltonian systems related by linear canonical transformations, which are combinations of the scale and gauge transformations, are considered. Using path integrals, it is proven that each quantum Hamiltonian can be expressed by the classical Hamiltonian whose canonical variables are replaced by their corresponding quantum operators. The relation between the Schr\"odinger solution and the propagator for the transformed (new) and for the original (old) systems are evaluated by applying the unitary operator which describes the linear relationship between their quantum operators as the corresponding relations between their classical forms. It is shown that the uncertainty relations between the canonical position and momentum operator depend on the gauge function chosen, and satisfy Heisenberg's uncertainty principle. The uncertainty relations between the canonical position and kinetic momentum operator (distinguished from the momentum operator) do not depend on the gauge function chosen, and may not satisfy Heisenberg's uncertainty principle. By the gauge transformation, a single system has innumerable Schr\"odinger equations, but the quantum averages of the function of the position and kinetic momentum operators are invariant for all solutions, as for classical cases.
A formulation of quantum mechanics in terms of complex canonical variables is presented. It is seen that these variables are governed by Hamilton's equations. It is shown that the action variables need to be quantized. By formulating a quantum Hamilton equation for the momentum variable, the energies for two different systems are determined. Quantum canonical transformation theory is introduced and the geometrical significance of a set of generalized quantization conditions which are obtained is discussed.
The nontrivial transformation of the phase space path integral measure under certain discretized analogs of canonical transformations is computed. This Jacobian is used to derive a quantum analog of the Hamilton-Jacobi equation for the generating function of a canonical transformation that maps any quantum system to a system with a vanishing Hamiltonian. A perturbative solution of the quantum Hamilton-Jacobi equation is given. This solution gives a new way to compute quantum corrections for any soliton equation for which action-angle variables are known.
The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.
In the last few years an extensive literature has developed on linear canonical tranformations and their representation in quantum mechanics. Applications of these results have been made to clustering theory in nuclei, problems of accidental degeneracy, etc. In the present paper we wish to turn our attention to nonlinear canonical transformations. We show that by dealing with appropriate functions fα (α=1,...,2n) of xi, pi (i=1,...,n) rather than with these variables themselves, we can in principle set unambiguously the equations that determine the representation in quantum mechanics of the canonical transformation under study. This result holds when the old and new functions fα have the same spectrum. We discuss specific examples when this last condition is satisfied: nonlinear canonical transformations in the radial variable that were obtained from projection of linear ones in higher-dimensional spaces; canonical transformations that take us from one Hamiltonian to another with the same spectrum, be this one continuous or discrete; canonical transformations that relate two sets of integrals of motion (which include the Hamiltonians) when we are dealing with phase spaces of dimensions higher than 2, etc. We discuss briefly, in the concluding section, the possibility of extending our analysis to canonical tranformations that do not conserve the spectrum of the relevant operators.