Optimized basis expansion as an extremely accurate technique for solving time-independent Schrödinger equation

Pouria Pedram 0 Mahdi Mirzaei 1 Siamak S Gousheh 1 0 Department of Physics, Science and Research Branch, Islamic Azad University , Tehran, Iran 1 Department of Physics, Shahid Beheshti University , Evin, Tehran 19839, Iran We use the optimized trigonometric finite basis method to find energy eigenvalues and eigenfunctions of the time-independent Schrdinger equation with high accuracy. We apply this method to the quartic anharmonic oscillator and the harmonic oscillator perturbed by a trigonometric anharmonic term as not exactly solvable cases and obtain the nearly exact solutions. - Background Eighty years after the birth of quantum mechanics, the Schrdingers famous equation still remains a subject of numerous studies, aiming at extending its field of applications and at developing more efficient analytic and approximation methods to obtain its solutions. There has always been a remarkable interest in studying exactly solvable Schrdinger equations. In this sense, the exact solubility has been found for only a very limited number of potentials, most of them being classified already by Infeld and Hull [1] on the basis of the Schrdinger factorization method [2-4]. However, a vast majority of the problems of physical interest do not fall in the above category, and we have to resort to approximation techniques. The need for such methods have stimulated the development of more sophisticated integration approaches such as embedded exponentially fitted Runge-Kutta [5], dissipative Numerov-type method [6], relaxational approach [7] based on the Henyey algorithm [8], an adaptive basis set using a hierarchical finite element method [9], and an approach based on microgenetic algorithm [10], which is a variation of a global optimization strategy proposed by Holland [11]. We can also mention the variational sinc collocation method [12] and the instanton method [13]. In this paper, we expand the wave function in terms of an orthonormal set of the eigenfunctions of a Hermitian operator, namely, the basis-set expansion method. Indeed, we use the trigonometric basis functions obeying periodic boundary condition. The accuracy of the solutions strongly depends on the domain of the wave function. So, we implement the Rayleigh-Ritz variational method to find the domains optimal value. The application of this method for the Dirichlet boundary condition is also discussed in [14,15]. Note that an analytic relation for this optimal length is recently presented in [16] which is only applicable for the power low potentials. A twodimensional application of this method is also discussed in the context of quantum cosmology [17]. The remainder of this paper is organized as follows. In The trigonometric basis-set expansion method Section, we present the underlying theoretical bases for the formulation of the trigonometric basis-set expansion method and the optimization procedures. In Section Applications, to illustrate the method, we apply this method for the simple harmonic oscillator (SHO). We then solve two perturbed harmonic oscillators that are not exactly solvable, the first with a quartic anharmonic term, and second with a rapidly oscillating trigonometric anharmonic term. We present our conclusions in the Conclusions Section. By multiplying both sides of the above equation by gi ( mLx ) and integrating over the x-space and using the orthonormality condition of the basis functions, one finds The trigonometric basis-set expansion method Let us consider the time-independent one-dimensional Schrdinger equation: f(x) = 2 V (x), As mentioned before, we use the trigonometric basis set to find the energy spectrum. Since we need to choose a finite subspace of a countably infinite basis, we restrict ourselves to the finite region L < x < L. This means that we can expand the solution as i=1 m=0 In the above choice of the basis, we are implicitly assuming periodic boundary condition. We can also make the following expansion: where Bm,i are coefficients that can be determined once f(x) is specified. By substituting Equations (4) and (6) into Equation (2) we obtain Because of the linear independence of gi( mLx ), every term in the summation must satisfy Am,i + Bm,i = Rm = D A = Bm,i = dx = Am ,i Cm,m ,i,i . L Therefore, we can rewrite Equation (8) as Cm,m ,i,i Am ,i = where the coefficients Cm,m ,i,i are defined by Equation (10). It is obvious that the presence of the operator f(x) in Equation (2) leads to nonzero coefficients Cm,m ,i,i in Equation (11), which in principle could couple all of the matrix elements of A. It is easy to see that the more basis functions we include, the closer our solution will be to the exact one. By selecting a finite subset of the basis functions, e.g., choosing the first 2N which could be accomplished by letting the index m run from 1 to N in the summations, Equation (11) can be written as where D is a square matrix with (2N ) (2N ) elements. The eigenvalues and eigenfunctions of the Schrdinger equation are approximately equal to the corresponding quantities of the matrix D, that is, the solution to this matrix equation simultaneously yields 2N sought after eigenstates and eigenvalues. We are free to adjust two parameters: 2N (the number of basis elements used) and 2L (the length of the spatial region). This length should be preferably larger than spatial spreading of all the soughtafter wave functions. However, if 2L is chosen to be too large, we loose overall accuracy. It is important to note that for each N, L has to be properly adjusted. This is in fact the optimization procedure, and we denote this optimal quantity by L(N ): for a few values of N, we compute ( N , L) which invariably has an inflection point in the periodic boundary condition. Therefore, all we have to do is to compute the position of this inflection point and compute an interpolating function for obtaining L(N ). Applications In this section, for illustrative purposes, we first apply the optimization procedure to find the bound states of a SHO which is an exactly solvable case. We then apply this method to two perturbed harmonic oscillators, the first with a quartic anharmonic term, and the second with a rapidly oscillating trigonometric anharmonic term. Table 1 The results for the first ten eigenvalues and eigenfunctions of the SHO with N = 100 n nexact E0 1.5 Figure 1 Ground state energy for SHO versus L for N = 5 in units where = 2. Simple harmonic oscillator The Schrdinger equation for SHO is The exact solutions are n(x) = 1/4 Hn(x) e x2/2, 2nn! n = n = 0, 1, 2, ..., where Hn(x) denote the Hermite polynomials. In Figure 1, we showed the ground state energy computed for N = 5 as a function of L using periodic boundary condition. Note the existence of the inflection point that determines L(5). We repeat this procedure for Figure 2 L versus N and its interpolating function. Figure 3 Exact and approximate ground state wave functions of SHO for N = 1, 2 with optimized L = {2.5 (...truncated)