Solution of the Heisenberg equations of motion for a double-well potential: Matrix elements and the energy eigenvalues

Boguslaw Stec, Czesław Jędrzejek

LI Feng-min

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.

A Acus, A Dargys

Analytical expressions for spectrum, eigenfunctions and dipole matrix elements of a square double quantum well (DQW) are presented for a general case when the potential in different regions of the DQW has different heights and effective masses are different. This was achieved by Gröbner basis algorithm which allows to disentangle the resulting coupled polynomials without explicitly solving the transcendental eigenvalue equation.

O. I. Ivanova, R. Kh. Sabirov

Olavi Hellman

Francis R. Halpern

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 ${x}^{4}$ 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.