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College Physics

Matrix elements and classical limits of nonlinear oscillator

LI Feng-min

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

For a nonlinear harmonic oscillator with potential V(x)=1/2mω2x2+λx4,the classical equation of motion can not be solved by the perturbation method.The classical solution of the nonlinear oscillator is derived from quantum matrix elements by using the Heisenberg correspondence principle,which helps us to understand the properties of the nonlinear oscillator more profoundly.

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For a nonlinear harmonic oscillator with potential V(x)=1/2mω2x2+λx4,the classical equation of motion can not be solved by the perturbation method.The classical solution of the nonlinear oscillator is derived from quantum matrix elements by using the Heisenberg correspondence principle,which helps us to understand the properties of the nonlinear oscillator more profoundly.
Heisenberg's form of the correspondence principle for non-relativistic matrix elements has been used to evaluate matrix elements for various potentials. These are compared with available quantum mechanical results to check the general validity of the use of classical mechanics and correspondence principles in the derivation of quantum mechanical expressions. Using a one dimensional harmonic oscillator potential matrix elements of the form (n mod qm mod n+s) and (n mod pm mod n+s) are worked out. A Morse potential is then considered and the matrix elements for position, momentum, and kinetic energy are evaluated. Using a Coulomb potential a similar procedure is used for functions of various position coordinates, from which dipole and quadrupole moments from transitions between n, l states are calculated. The agreement with quantum mechanics is found to be generally good, and in some cases identical results are obtained.
Finite freedom degree harmonic oscillator system is studied as follows: To begin with, its equation of motion is given and its solution is obtained by solving the equation, then the Lagrange and Hamiltan equations of motion of harmonic oscillator system are derived from variational principle. Further more, after harmonic oscillator system are quantized, we can also get its motion equation stem from the Heisenbergs Equation. In addition, classical mechanics and quantum mechanics are simultaneously applied to many kinds of conditions (including n dimensional states) of harmonic oscillator systems. Finally, the equivalence of canonical equation and Heisenbergs Equation to harmonic oscillator systems are verified.
For a class of problems it is shown that a solution to Newton's equation of motion may be interpreted as the generating function for the matrix elements of the coordinate operators in the Heisenberg representation. For three one-dimensional problems, the harmonic oscillator, the particle in a box, and the x4{x}^{4}x4 anharmonic oscillator, solutions to the quantum-mechanical problem are given in terms of the classical solution. A brief examination is made of a two-dimensional problem in which the angular momentum is conserved.
The m, n matrix element of an arbitrary potential function V(q) in the one-dimensional harmonic-oscillator representation is shown to be given by 〈m | V(q) | n〉= ∑ r=0mα(m!n!)12r!(m−r)!(n−r)! ∫ −∞∞dyg(αy)e−12y2(iy)m+n−2r,where α≡(2ω/ℏ)½, and g(αy) is the Fourier transform of V(q). This formula is specialized to the cases where V(q) is given by qj, qje−½γq2, ejαzq, and q−1 sin(αλq), where j is a nonnegative integer and γ, z, and λ are real parameters. Results are compared, where possible, with previous work.
Matrices of dynamical variables of the one-dimensional harmonic oscillator are calculated by a new semiclassical method formulated by More and Warren. The results are semiclassical matrices Xn, m, Pn, m, etc., which i) exactly obey selection rules for zero matrix-elements, ii) are very accurate, circa 1 %, for nonzero matrix-elements, iii) which diagonalize the Hamiltonian H, and iv) which exactly satisfy the Heisenberg commutation relations and equations of motion. These statements are true for matrix-elements Fn, m with m ≽ n. The results show that most of the quantum physics is contained in the extended semiclassical theory.