arXiv subjects
Publications and source records attributed to W. Q. Zhao.
An explicit convergent iterative solution for the lowest energy state of the Schroedinger equation with generalized $N$-dimensional Sombrero-shaped potential is presented. The condition for the convergence of the iteration procedure and the dependence of the shape of the groundstate wave function on the parameters are discussed.
We present an explicit convergent iterative solution for the lowest energy state of the Schroedinger equation with a generalized double well potential $V=\frac{g^2}{2}(x^2-1)^2(x^2+a)$. The condition for the convergence of the iteration procedure and the dependence of the shape of the groundstate wave function on the parameter $a$ are discussed.
The convergent iterative procedure for solving the groundstate Schroedinger equation is extended to derive the excitation energy and the wave function of the low-lying excited states. The method is applied to the one-dimensional quartic potential problem. The results show that the iterative solution converges rapidly when the coupling $g$ is not too small.
We present an explicit convergent iterative solution for the lowest energy state of the Schroedinger equation with an $N$-dimensional radial potential $V=\frac{g^2}{2}(r^2-1)^2$ and an angular momentum $l$. For $g$ large, the rate of convergence is similar to a power series in $g^{-1}$.
The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potentials. As an example, the method is applied to discuss the ground state solution of Yukawa potential, using Hulthen solution as the trial function.
We present a new convergent iterative solution for the two lowest quantum wave functions $ψ_{ev}$ and $ψ_{od}$ of the Hamiltonian with a quartic double well potential $V$ in one dimension. By starting from a trial function, which is by itself the exact lowest even or odd eigenstate of a different Hamiltonian with a modified potential $V+δV$, we construct the Green's function for the modified potential. The true wave functions, $ψ_{ev}$ or $ψ_{od}$, then satisfies a linear inhomogeneous integral equation, in which the inhomogeneous term is the trial function, and the kernel is the product of the Green's function times the sum of $δV$, the potential difference, and the corresponding energy shift. By iterating this equation we obtain successive approximations to the true wave function; furthermore, the approximate energy shift is also adjusted at each iteration so that the approximate wave function is well behaved everywhere. We are able to prove that this iterative procedure converges for both the energy and the wave function at all $x$.
We present a new method to derive low-lying N-dimensional quantum wave functions by quadrature along a single trajectory. The N-dimensional Schroedinger equation is cast into a series of readily integrable first order ordinary differential equations. Our approach resembles the familiar W.K.B. approximation in one dimension, but is designed to explore the classically forbidden region and has a much wider applicability than W.K.B.. The method also provides a perturbation series expansion and the Green's functions of the wave equation in N-dimension, all by quadratures along a single trajectory. A number of examples are given for illustration, including a simple algorithm to evaluate the Stark effect in closed form to any finite order of the electric field.
We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with -V as its potential. The function V is $\geq 0$, and can have several minina (V=0). We assume the problem to be characterized by a small anhamornicity parameter $g^{-1}$ and a much smaller quantum tunneling parameter $ε$ between these different minima. Expanding either the wave function or its energy as a formal double power series in $g^{-1}$ and $ε$, we show how the coefficients of $g^{-m}ε^n$ in such an expansion can be expressed in terms of definite integrals, with leading order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potential $V={1/2}g^2(x^2-a^2)^2$.