Wave Equations for Simple Quantum System

Xiang-Yao Wu, Baijun Zhang, Haibo Li +3 more

Donald H. Kobe

The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schroedinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton's equations also give the Schroedinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order derivatives. For a time-independent system the Principle of Least Action reduces to the energy variational principle. For time-dependent systems the Principle of Least Action gives time-dependent approximations. Using a Hartree product trial wave function for a time-dependent many-boson system, we apply the Principle of Least Action to obtain the Gross-Pitaevskii equation that describes a Bose-Einstein condensate.

Eric R. Lyon

It can be shown that, for continuity of u and u' at xl, k = 27r/8 = 45 deg. The surface through xl, serving as a boundary between the regions occupied by the exponential and oscillatory solutions, is the well-known potential barrier. There is an equivalence between much of the W-K-B method and the following simpler set of equations. In place of the time-eliminated wave equation of optics, write (d2u/dx2) sin2a + (I/x) (du/dx) cos2a + (47r2/X2)U = O, which becomes the time-eliminated wave equation of optics in one dimension when a = 90 deg. Solutions: u' -A exp (-27r i tanr 0.34657),. . . x , ... where cosa = I/x, sina (x2-X2) 1/2/x, tana = (x2-X2) 1//X . When x = -, u'= 0.7071 A =A sin 27r/8 = u. It may be shown that du'/dx = d'u/dx5 when x = ?+ . The surface through x = -X, serving as a boundary between the regions occupied by the exponential and oscillatory solutions, is, therefore, equivalent to the W-K-B potential barrier. In fact, the above simple equations may be given the form of the W-K-B equations by the same substitutions which transform the wave equation of optics into the Schr6dinger equation. When a consistent vector system of directions is assigned to u, x and X, the wave motion, described by the above simple equations, has conical wave fronts.

Sergey K. Ivanov, A. M. Kamchatnov

We study the dynamics of so-called simple waves in a two-component Bose-Einstein condensate. The evolution of the condensate is described by Gross-Pitaevskii equations which can be reduced for these simple wave solutions to a system of ordinary differential equations which coincide with those derived by Ovsyannikov for the two-layer fluid dynamics. We solve the Ovsyannikov system for two typical situations of large and small difference between interspecies and intraspecies nonlinear interaction constants. Our analytic results are confirmed by numerical simulations.