Solution of Free-Particle Radial Dependant Schrödinger Equation Using He's Homotopy Perturbation Method

Homotopy perturbation method (HPM) is one of the newest analytical methods to solve linear and nonlinear differential equations. In this paper, HPM is used to formulate a new analytic solution of free-particle radial dependent Schrödinger equation. In contrast to the traditional perturbation methods, the homotopy method does not require a small parameter in the equation. In this method, according to the homotopy technique, a homotopy with an embedding parameter [0,1] δ ∈ is constructed, and the embedding parameter is considered as a small parameter. The obtained result shows the evidence of simplicity, usefulness, and effectiveness of the homotopy perturbation method for obtaining approximate analytical solutions for the radial dependent Schrödinger equation.


Introduction
In order to solve radial dependant Schrödinger equation for a free-particle; V(r) = 0, we will examine the application of the Homotopy Perturbation Method (HPM), which was proposed first by He (1999).The HPM is designed for solving differential and integral equations, linear and nonlinear, and has been the subject of extensive analytical and numerical studies.The method, which is a coupling of a homotopy technique and a perturbation technique, deforms continuously to a simple problem which is easily solved.This method, which does not require a small parameter in an equation, In contrast to the traditional perturbation methods, has a significant advantage in that it provides an analytical approximate solution to a wide range of nonlinear problems in applied sciences.This method doesn't need linearization, perturbation or un-justified assumptions.The HPM yields the solution in terms of a rapid convergent series with easily computable components (He, 2003).
In the last two decades with the rapid development of differential equations science, there has appeared ever-increasing interest of scientists and engineers in the analytical techniques for linear and nonlinear problems.The widely applied techniques are perturbation methods.Latif (2005) applied the HPM to search for exact analytical solutions of linear differential equations with constant coefficients.In addition, based on the precise integration method, a coupling technique of the variational iteration method (VIM) and HPM is proposed to solve nonlinear matrix differential equations.Rezania et al (2009) used HPM and VIM to solve the heat equations which are functions on time and space.This type of equation governs on numerous scientific and engineering experimentations.Zhang et al (2006) obtained an explicit analytical solution for nonlinear Poisson-Boltzmann equation by the HPM.Wang et al (2007) applied HPM to solve reaction-diffusion equations which is governed by the nonlinear ordinary differential equation.Furthermore, HPM is also applied to solve the Helmholtz equation, and the results reveal that this method is very effective and simple (Bizzar et al, 2008).In this work, HPM, in a realistic and efficient way, is proposed to provide approximate solutions for Free-particle radial dependent Schrödinger equation (spherical Bessel equation).

Basic Idea of HPM:
To illustrate the basic ideas of the new HPM, we consider the following nonlinear differential equation where A is a general differential operator , F(r) is a known analytic function , Г is the boundary of the domain Ω .The operator A can , generally speaking, be divided into two parts L and N , where L is linear, while N is nonlinear.Eq. ( 1), therefore, can be written as follows (He, 2003): (3) By the homotopy technique (Liao, 1997), we construct a homotopy v (r, δ): (5) Where δ ∈ [0, 1] is an embedding parameter, u o is an initial approximation of Eq. ( 1), which satisfies the boundary conditions.Obviously, from Eq. ( 4) and Eq. ( 5), we have The changing process of δ from zero to unity is just that of v(r, δ) from u o (r) to u(r).In topology, this is called deformation, and L(v) -L(u o ), A(v) -F(r) are called homotopic.We use the imbedding parameter δ as a "small parameter", and assume that the solution of Eq. ( 5) can be written as a power series in δ: ) Setting δ = 1 result in approximate solution of Eq. (1): The coupling of the perturbation method is called the homotopy perturbation method, which has eliminated limitations of traditional methods.In the other hand, the proposed technique can take full advantage of the traditional perturbation techniques (He, 2004).The series ( 9) is convergent for most cases, however, the convergent rate depends upon the nonlinear operator A(v): 1) The second derivative of N (v) with respect to v must be small, because the parameter δ may be relatively large, i.e. δ → 1.
2) The norm of must be smaller than one, in order that the series converges.

Free-Particle Radial Schrödinger Equation
In spherical coordinates, the Laplacian takes the form The time-independent Schrödinger equation reads ( 11) Where ˆo H is the Hamiltonian, V(r) is the potential energy, and E is the energy of the particle.In spherical coordinates, the Hamiltonian ˆo For a free particle, V(r) = 0, then ˆo The first pat which involves only derivative with respect to r, describes the kinetic energy of the radial motion.
The second part, describes the kinetic energy of the angular motion.The quantum angular momentum of the particle is given by the operator (Schiff, 1968) (14) where ∇ is the expression for gradient in spherical coordinate and has the form But and hence the unit vectors θ and φ are resolved into their Cartesian components (Tannoudji, 1977) sin cos cot cos sin cot Therefore, the partial differential operator takes the form and this operator involves only derivatives with respect to θ and ϕ .Hence, it is easy to rewrite the free particle Hamiltonian of Eq. ( 13), which becomes L belonging to same eigenvalue (Dirac, 1999).In order to solve the Schrödinger which is called the free radial Schrödinger operator.The angular kinetic energy appears in the form of a potential energy 2 ( 1)/r + l l .This term is called centrifugal potential energy or centrifugal barrier, because it has the effect of a repulsive force in the radial direction (Flugge, 1971).Immediately it can be shown that ˆo After multiplying by r 2 and rearranging the terms, we get where 2 2 2 / k mE = h , therefore Eq. ( 25) becomes ( ) Substitution of Eq. ( 27) and Eq. ( 28) into Eq.( 26) gives ( ) Now, we look for a solution of the form Substitution of Eq. ( 30) and Eq. ( 32) into Eq.( 26) leads to ( ) If we multiply Eq. ( 33) by 1/ 2 ξ − , we will get ( ) Summing up similar terms yields ( ) But the solutions to this equation are the ordinary Bessel functions of half integral order; are related to the spherical Bessel and Neumann functions, j l and n l , respectively by: The Bessel functions j l are finite at the origin, but the Neumann functions blows up at the origin.Accordingly, we must have 0 B = l , and hence the normalized solutions of Eq. ( 26) are In this section, we implement the HPM, in a realistic and efficient way, to provide approximate solutions for the radial dependence Schrödinger equation (spherical Bessel equation) subject to the condition that R(r) is bounded at r = 0.For the sake of continuity, this equation is rewritten here ( ) (42) In view of Eq. ( 4) or ( 5), the homotopy for Eq. ( 42) can be constructed as The basic assumption of HPM is that the solution R(r) can be expressed as a power of series in δ.
The terms up to δ 3 are considered, where where, For computing purposes, the solution provided by Eq. ( 61) as R(r) can be approximated by the p th term R p (r)

Conclusion
As a result, we conclude that the HPM can provides the expected solution and works successfully in handling differential equations directly which produces the solutions in terms of convergent series with easily computable components, and requires less computational work when compared with other methods.A convergent series solution for the radial dependent Schrödinger equation has been obtained, which is realistic and effective.
the radial kinetic energy and hence with ˆo H .The kinetic energy operator also commutes with ˆz L , because ˆz L commutes with 2 L and with the expression depending on r.This proves that the operator of the kinetic energy is invariant under rotations (Eisberg and Rinsik, 1985): (21) The same commutation relation holds for the operators in Cartesian coordinates, which are related to the operator in spherical coordinate.Any eigenspace of 2 L is left invariant by ˆo H .If ψ l is an eigenvector of 2 L , then ˆo H ψ l is an eigenvector of 2 eigen-functions of 2 L with eigen-values 2 ( 1) + h l l .We see immediately that the partial differential operator of ˆo H becomes an ordinary differential operator R(r) up to the third term is illustrated in Fig. (3).

Figure 1 .
Figure 1.The radial solution R 1 (r) of Radial dependent Schrödinger equation (spherical Bessel equation) at different values of l .

Figure 2 .
Figure 2. The radial solution R 2 (r) of the radial dependent Schrödinger equation (spherical Bessel equation) at different values of l .

Figure 3 .
Figure 3.The general solution R(r) of the radial dependent Schrödinger equation (spherical Bessel equation) up to the third term of Eq. (61), at different values of l .