The main differential equations of quantum theory are the eigenequations based on the energy operator;they have the energy as eigenvalues and the wave functions as eigenfunctions. A usual complexity of these equations...The main differential equations of quantum theory are the eigenequations based on the energy operator;they have the energy as eigenvalues and the wave functions as eigenfunctions. A usual complexity of these equations makes their accurate solutions accessible easily only for very few physical cases. One of the methods giving the approximate solutions is the Schrödinger perturbation theory in which both the energies and wave functions of a more complicated eigenproblem are approached with the aid of similar parameters characteristic for a less complicated eigenproblem. No time parameter is necessary to be involved in these calculations. The present paper shows that the Schrödinger perturbation method for non-degenerate stationary quantum states, i.e. the states being independent of time, can be substantially simplified by applying a circular scale of time separately for each order of the perturbation theory. The arrangement of the time points on the scale, combined with the points contractions, gives almost immediately the series of terms necessary to express the stationary perturbation energy of a given eigenproblem. The Schrödinger’s method is compared with the Born-Heisenberg-Jordan perturbation approach.展开更多
Using the method of separation of variables in the elliptical coordinate system, a recursive formula for the electromagnetic fields in a confocal elliptical waveguide filled with multi-layered homogeneous isotropic me...Using the method of separation of variables in the elliptical coordinate system, a recursive formula for the electromagnetic fields in a confocal elliptical waveguide filled with multi-layered homogeneous isotropic media is derived; then the eigenequation for it is given. When an elliptical waveguide becomes a circular waveguide, the electromagnetic fields and the eigenequation of the circular waveguide can be obtained from the eigenequation of the elliptical waveguide using the asymptotic formulae of Mathieu and modified Mathieu functions for a large radial coordinate in the elliptical coordinate system, and the eigenequation of a circular waveguide filled with multilayered dielectrics can be treated as a special case of an elliptical waveguide. In addition, some numerical examples are presented to analyze the propagating characteristics influenced by the permittivity, permeability of dielectrics filled in the elliptical waveguide, etc. The results show that changing the permittivity or permeability of the dielectrics filled in the waveguide and the major semiaxis value of the i-th layer can change the propagating characteristics of an elliptical waveguide.展开更多
In this paper an orthogonal function method is presented based on the idea to suppose perioche sohuion with the method of harmonie balance The displaeement is expressed in the form of trigonometric fumctions a group o...In this paper an orthogonal function method is presented based on the idea to suppose perioche sohuion with the method of harmonie balance The displaeement is expressed in the form of trigonometric fumctions a group of simplified digenequationsare obtained by the use of orthogonarity of trigonometric fumetions and linear mondes The method overcomes the diffieulty of a drifi term existing in systems with quadratic nonlinearities .The ealeulation examples show that the method has thd advantages of high caleulation preeision high convergenee speed and littld ealeulation work展开更多
文摘The main differential equations of quantum theory are the eigenequations based on the energy operator;they have the energy as eigenvalues and the wave functions as eigenfunctions. A usual complexity of these equations makes their accurate solutions accessible easily only for very few physical cases. One of the methods giving the approximate solutions is the Schrödinger perturbation theory in which both the energies and wave functions of a more complicated eigenproblem are approached with the aid of similar parameters characteristic for a less complicated eigenproblem. No time parameter is necessary to be involved in these calculations. The present paper shows that the Schrödinger perturbation method for non-degenerate stationary quantum states, i.e. the states being independent of time, can be substantially simplified by applying a circular scale of time separately for each order of the perturbation theory. The arrangement of the time points on the scale, combined with the points contractions, gives almost immediately the series of terms necessary to express the stationary perturbation energy of a given eigenproblem. The Schrödinger’s method is compared with the Born-Heisenberg-Jordan perturbation approach.
文摘Using the method of separation of variables in the elliptical coordinate system, a recursive formula for the electromagnetic fields in a confocal elliptical waveguide filled with multi-layered homogeneous isotropic media is derived; then the eigenequation for it is given. When an elliptical waveguide becomes a circular waveguide, the electromagnetic fields and the eigenequation of the circular waveguide can be obtained from the eigenequation of the elliptical waveguide using the asymptotic formulae of Mathieu and modified Mathieu functions for a large radial coordinate in the elliptical coordinate system, and the eigenequation of a circular waveguide filled with multilayered dielectrics can be treated as a special case of an elliptical waveguide. In addition, some numerical examples are presented to analyze the propagating characteristics influenced by the permittivity, permeability of dielectrics filled in the elliptical waveguide, etc. The results show that changing the permittivity or permeability of the dielectrics filled in the waveguide and the major semiaxis value of the i-th layer can change the propagating characteristics of an elliptical waveguide.
文摘In this paper an orthogonal function method is presented based on the idea to suppose perioche sohuion with the method of harmonie balance The displaeement is expressed in the form of trigonometric fumctions a group of simplified digenequationsare obtained by the use of orthogonarity of trigonometric fumetions and linear mondes The method overcomes the diffieulty of a drifi term existing in systems with quadratic nonlinearities .The ealeulation examples show that the method has thd advantages of high caleulation preeision high convergenee speed and littld ealeulation work
