Calculation of the eigenfunctions and eigenvalues of Schrödinger type equations by asymptotic Taylor expansion method (ATEM)
aa r X i v : . [ m a t h - ph ] A ug Calculation of the eigenfunctions and eigenvalues of Schr¨odingertype equations by asymptotic Taylor expansion method (ATEM)
Ramazan Ko¸c ∗ and Eser Ol˘gar † Gaziantep University, Department of Physics,Faculty of Engineering 27310 Gaziantep/Turkey (Dated: August 5, 2010)
Abstract
A novel method is proposed to determine an analytical expression for eigenfunctions and numer-ical result for eigenvalues of the Schr¨odinger type equations, within the context of Taylor expansionof a function. Optimal truncation of the Taylor series gives a best possible analytical expressionfor eigenfunctions and numerical result for eigenvalues.
PACS numbers: 03.65.Ge; 03.65.Fd,02.30.-fKeywords: Asymptotic expansion method, Taylor series, Analytical solution, Schr¨odinger equation, Functiontheory ∗ Corresponding Author: E-mail address: [email protected], Phone: +90 342 317 2209, Fax: +90 342 36018 22 † E-mail address: [email protected] . INTRODUCTION One of the source of progress of the sciences depends on the study of the same problemfrom different point of view. Besides their progress of the sciences those different pointof views include a lot of mathematical tastes. Over the years considerable attention hasbeen paid to the solution of Schr¨odinger equation. Determination of eigenvalues of theSchr¨odinger equation via asymptotic iteration method (AIM) has recently attracted someinterest, arising from the development of fast computers [1–4]. This method have beenwidely applied to establish eigenvalues of the Schr¨odinger type equations [5–10]. Althoughthe AIM formalism is very efficient to obtain eigenvalues of the Schr¨odinger equation, itrequires tedious calculations in order to determine wave function if the system is not exactlysolvable. When Schr¨odinger equation includes a non solvable potential, the calculation ofwave function involved with a large number of terms will lose its simplicity and accuracy.In this paper we will discuss new formalism based on the Taylor series expansion method,namely Asymptotic Taylor Expansion Method (ATEM). Although, Taylor Series Method[11] is an old one, it appears, however, not has been fully exploited in the analysis of bothin solution of physical and mathematical problems. Yet, even today, new contributions tothis problem are being made [12]. Apart from its formal relation to AIM, ATEM has alsobeen easily applied to solve second order linear differential equations by introducing a simplecomputer program. We would like to mention here that the ATEM is a field of tremendousscope and has an almost unlimited opportunity, for its applications in the solution of theSchr¨odinger type equations. One can display a number of fruitful applications of the ATEMin different fields of the physics. For instance, our formalism of ATEM gives a new approachto the series solution of the differential equations as well as interrelation between seriessolution of differential equations and AIM. The method can also be applied to solve Diracequation and Klein Gordon equation. In this paper we address ourselves to the solution ofthe eigenvalue problems by using the ATEM.One of the fundamental advantage of ATEM is that (approximate) analytical expressionsfor the wave function of the associated Hamiltonian can easily be obtained. We note thatATEM also gives an accurate result for the eigenvalues when it is compared to AIM.It should also be noted here that the determination of wave function by using AIM requiretedious calculations if the system is not exactly solvable. For non exactly solvable potentials,2he calculation of wave function by using AIM involved with a large number of terms will loseits simplicity and accuracy. Therefore, the method introduced here is useful to determinean analytical expression for the wave function of the non exactly solvable equations.The paper is organized as follows. The first main result of the paper is given in section byreformulating the well known Taylor series expansion of a function. Section 3 is devoted tothe application of the main result for solving the Schr¨odinger equation including various po-tentials. As a practical example, we illustrate solution of the Schr¨odinger equation includinganharmonic oscillator potential and the Hamiltonian of an interacting electron in a quantumdot. In this section we present an approximate analytical expression for eigenfunction andnumerical results for eigenvalues of the anharmonic oscillator potential and the Hamiltonianof an interacting electron in a quantum dot. We also analyze asymptotic behavior of theHamiltonian. Finally we comment on the validity of our method and remark on its possibleuse in different fields of the physics in section 4.
II. A NEW FORMALISM OF THE TAYLOR EXPANSION METHOD
In this section, we show the solution of the Schr¨odinger equation for a quite ample classof potentials, by modifying Taylor series expansion by means of a finite sequence insteadof an infinite sequence and its termination possessing the property of quantum mechanicalwave function. In quantum mechanics bound state energy of the atom is quantized andeigenvalues are discrete and for each eigenvalues there exist one or more an eigenfunctions.When we are dealing with the solution of the Schr¨odinger equation we are mainly interestedin the discrete eigenvalues of the problem. The first main result of this conclusion givesnecessary and sufficient conditions for the termination of the Taylor series expansion of thewave function.Let us consider Taylor series expansion of a function f ( x ) about the point a : f ( x ) = f ( a ) + ( x − a ) f ′ ( a ) + 12 ( x − a ) f ′′ ( a ) + 16 ( x − a ) f (3) ( a ) + · · · ∞ = X n =0 ( x − a ) n n ! f ( n ) ( a ) (1)where f ( n ) ( a ) is the n th derivative of the function at a . Taylor series specifies the value ofa function at one point, x , in terms of the value of the function and its derivatives at a3eference point a . Expansion of the function f ( x ) about the origin ( a = 0), is known asMaclaurin’s series and it is given by, f ( x ) = f (0) + xf ′ (0) + 12 x f ′′ (0) + 16 x f (3) (0) + · · · ∞ = X n =0 x n n ! f ( n ) (0) . (2)Here we develop a method to solve a second order linear differential equation of the form: f ′′ ( x ) = p ( x ) f ′ ( x ) + q ( x ) f ( x ) . (3)It is obvious that the higher order derivatives of the f ( x ) can be obtained in terms of the f ( x ) and f ′ ( x ) by differentiating (3). Then, higher order derivatives of f ( x ) are given by f ( n +2) ( x ) = p n ( x ) f ′ ( x ) + q n ( x ) f ( x ) (4)where p n ( x ) = p ( x ) p n − ( x ) + p ′ n − ( x ) + q n − ( x ) , and q n ( x ) = q ( x ) p n − ( x ) + q ′ n − ( x ) . (5)Of course, the last result shows there exist a formal relation between AIM and ATEM. To thisend, we conclude that the recurrence relations (5) allow us algebraic exact or approximateanalytical solution of (3) under some certain conditions. Let us substitute (5) into the (1) toobtain the function that is related to the wave function of the corresponding Hamiltonian: f ( x ) = f (0) m X n =2 q n − (0) x n n ! ! + f ′ (0) m X n =2 p n − (0) x n n ! ! . (6)After all we have obtained an useful formalism of the Taylor expansion method. This formof the Taylor series can also be used to obtain series solution of the second order differentialequations. In the solution of the eigenvalue problems, truncation of the the asymptoticexpansion to a finite number of terms is useful. If the series optimally truncated at thesmallest term then the asymptotic expansion of series is known as superasymptotic [13], andit leads to the determination of eigenvalues with minimum error.Arrangement of the boundary conditions for different problems becomes very importantbecause improper sets of boundary conditions may produce nonphysical results. When onlyodd or even power of x collected as coefficients of f (0) or f ′ (0) and vice verse, the series4s truncated at n = m then an immediate practical consequence of these condition for q m − (0) = 0 or p m − (0) = 0 . In this way, the series truncates at n = m and one of theparameter in the q m − (0) or p m − (0) belongs to the spectrum of the Schr¨odinger equation.Therefore eigenfunction of the equation becomes a polynomial of degree m . Otherwisethe spectrum of the system can be obtained as follows: In a quantum mechanical systemeigenfunction of the system is discrete. Therefore in order to terminate the eigenfunction f ( x ) we can concisely write that q m (0) f (0) + p m (0) f ′ (0) = 0 q m − (0) f (0) + p m − (0) f ′ (0) = 0 (7)eliminating f (0) and f ′ (0) we obtain q m (0) p m − (0) − p m (0) q m − (0) = 0 (8)again one of the parameter in the equation related to the eigenvalues of the problem.We can state that the ATEM reproduces exact solutions to many exactly solvable differ-ential equations and these equations can be related to the Schr¨odinger equation. It will beshown in the following section ATEM also gives accurate results for non-solvable Schr¨odingerequations, such as the sextic oscillator, cubic oscillator, deformed Coulomb potential, etc.which are important in applications to many problems in physics. This asymptotic ap-proach opens the way to the treatment of Schr¨odinger type equation including large classof potentials of practical interest. III. SOLUTION OF THE SCHR ¨ODINGER EQUATION BY USING ATEM
An analytical solution of the Schr¨odinger equation is of high importance in nonrelativisticquantum mechanics, because the wave function contains all necessary information for fulldescription of a quantum system. In this section we take a new look at the solution of theSchr¨odinger equation by using the method of ATEM developed in the previous section. Letus consider the following eigenvalue problem (¯ h = 2 m = 1) − d ψ ( x ) dx + V ( x ) ψ ( x ) = Eψ ( x ) (9)where V ( x ) is the potential, ψ ( x ) is wave function and E is the energy of the system. Theequation has been solved exactly for a large number of potentials by employing various tech-5iques. In general, it is difficult to determine the asymptotic behavior of (9) in the presentform. Therefore it is worthwhile to transform (9) to an appropriate form by introducingthe wave function ψ ( x ) = f ( x ) exp (cid:18) − Z W ( x ) dx (cid:19) . Thus, this change of wave functionguaranties lim x − > ∞ ψ ( x ) = 0. We recast (9) and we obtain the following equation L ( x ) = − f ′′ ( x ) + 2 W ( x ) f ′ ( x ) + ( V ( x ) + W ′ ( x ) − W ( x ) − E ) f ( x ) = 0 . (10)In this formalism of the equation coefficients in (3) can be expressed as: p ( x ) = 2 W ( x ) , q ( x ) = ( V ( x ) + W ′ ( x ) − W ( x ) − E ) . Using the relation given in (5) one can easily compute p n ( x ) and q n ( x ) by a simple MATH-EMATICA program. Our task is now to illustrate the use of ATEM to obtain explicitanalytical solution of the Schr¨odinger equation including various potentials.
1. Anharmonic oscillator
Solution of the Schr¨odinger equation including anharmonic potential has attracted a lotof attention, arising its considerable impact on the various branches of physics as well asbiology and chemistry. The equation is described by the Hamiltonian H = − d dx + x + gx . (11)In practice anharmonic oscillator problem is always used to test accuracy and efficiencyof the unperturbative methods. Let us introduce, the asymptotic solutions of anharmonicoscillator Hamiltonian when W ( x ) = x , then the wave function takes the form ψ = e − x f ( x ) , and (10 )can be expressed as L ( x ) = − d fdx + 2 x dfdx + ( gx + 1 − E ) f = 0 . (12)Comparing the equations (3) and (12) we can deduce that p ( x ) = 2 x and q ( x ) = ( gx + 1 − E ) . (13)6ere we take a new look at the solution of the (12) by using the method of ATEM developedin the previous section. By applying (8), the corresponding energy eigenvalues are calculatedby the aid of a MATHEMATICA program.The term asymptotic means the function approaching to a given value as the iter-ation number tends to infinity. By the aid of MATHEMATICA program we calcu-late eigenvalues E and eigenfunction f ( x ) for g = 0 . k = { , , , , , , } . The eigenvalues are presented in Table I and and are comparedwith results computed by the AIM [1] and direct numerical integration method [14] by taking g = 0 . . k n = 0 n = 1 n = 2 n = 3 n = 4 n = 520 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E [1] 1 . . . . . . E [14] 1 . . . . . . k and g = 0 . . Last two rowscorresponds the comparison of eigenvalues computed by the AIM [1], direct numerical integrationmethod [14] .
The function f ( x ) for n = 2 state is given in (14).7 IG. 1: The first four wavefunctions for anharmonic oscillator. k = 30; f ( x ) = 1 − . x + 0 . x + 0 . x − . × − x − . × − x + 8 . × − x k = 50; f ( x ) = 1 − . x + 0 . x + 0 . x − . × − x − . × − x + 8 . × − x (14) k = 80; f ( x ) = 1 − . x + 0 . x + 0 . x − . × − x − . × − x + 8 . × − x As we mentioned before using ATEM we can obtain an analytical expression for the wavefunction of the Schr¨odinger equation. Substituting E into (6) we get the wave function ofthe Schr¨odinger equation for the corresponding eigenvalues. Analytical expressions signalthat ATEM produce an efficient result for the eigenfunction. For the first four states theplot of the normalized wave functions are given in Figure 1.8 . Interacting electrons in a quantum dot In this section we present a procedure to solve the Schr¨odinger equation of two interactingelectrons in a quantum dot in the presence of an external magnetic field by using ATEM. Theproblem has been discussed in various articles [15–17]. Here we just solve the mathematicalpart of the problem. Without further discussion the Schr¨odinger equation for a quantumdot containing two electrons in the presence of the magnetic field B perpendicular to thedot is given by H = X i =1 ( 12 m ∗ i ( P i + eA ( r i )) + 12 m ∗ i ω r i ) + e ε | r − r | (15)Introducing relative and center of mass coordinates r = r − r , R = ( r + r ) theHamiltonian can be separated into two parts such that H = 2 H r + H R , where H r = p m ∗ + 12 m ∗ ω r + e ǫr + 12 ω c L r , (16a) H R = P m ∗ + 12 m ∗ ω R + 12 ω c L R . (16b)Equation (16b) is the Hamiltonian of the harmonic oscillator, and it can be solved exactly.Let us turn our attention to the solution of the Hamiltonian H r . In the polar coordinate r = ( r, α ), if the eigenfunction φ = r − e iℓα u ( r ) (17)is introduced, the Schr¨odinger equation H r φ = E r φ , can be expressed as (cid:18) − ¯ h m ∗ d dr + ¯ h m ∗ ( ℓ −
14 ) 1 r + 12 m ∗ ω r + e ǫr + 12 ω c L r (cid:19) u ( r ) = E r u ( r ) . (18)From now on we restrict ourselves to the solution of Eq. (18). After changing the variable r → ¯ h √ m ∗ r and substituting u ( r ) = r ℓ + e − ¯ hω r f ( r ) , we obtain the following equation L ( r ) = − rf ′′ ( r ) + ( ωr − (2 ℓ + 1)) f ′ ( r ) − ( rE n + λ ) f ( r ) = 0 , (19)where E r = E n + ( | ℓ | + 1)¯ hω + 12 L r ω c , λ = − e ε , (20)for simplicity we have chosen that ¯ hω = ¯ h m ∗ = 1 . In this case the functions p ( r ) and q ( r )are given by p ( r ) = ωr − ℓ + 1 r and q ( r ) = − E n − λr . (21)9eanwhile we bring to mind that Hamiltonian (15) possesses a hidden symmetry. Thisimplies that the Hamiltonian is quasi-exactly solvable [15, 16]. Fortunately, quasi exactsolvability of the Hamiltonian gives us an opportunity to check accuracy of our result andto test our method. In order to obtain quasi exact solution of (19) we set in: E n = jω, where j = 1 , , .... and then the problem is exactly solvable when the following relation is satisfied: λ = n ± p ω (2 ℓ + 1) o ; j = 1 λ = n , ± p ω (4 ℓ + 3) o ; j = 2 λ = (cid:26) , ± q ω ( ℓ + 1) ± ω √
73 + 128 ℓ + 64 ℓ (cid:27) ; j = 3 · · · . Note that λ, ω and ℓ belong to the spectrum of the Hamiltonian. Therefore an accuracycheck for the ATEM can be made. We have tested ATEM and the result are given by λ = n ± p ω (2 ℓ + 1) o ; j = 1; E n = ω ; f ( r ) = 1 ∓ r ω ℓ + 1 rλ = n ± p ω (4 ℓ + 3) o ; j = 2; E n = 2 ω ; f ( r ) = 1 ∓ r ω (4 ℓ + 3)2 ℓ + 1 r + ω ℓ + 1 r . Consequently, we demonstrated that our approach is able to reproduce exact results forthe exactly solvable second order differential equations. Let us turn our attention to thecomplete solution of the (19). We have again used 80 iterations during the solution of theequation and controlled the stability of the eigenvalues. The results are given in Table II.We go back (6) to obtain the wave function of the equation of(19) for various values of E. Their plots are given in Figure 2.
IV. CONCLUSION
The basic features of our approach are to reformulate Taylor series expansion of a func-tion for obtaining both eigenvalues and eigenfunctions of the Schr¨odinger type equations.10 n = 0 n = 1 n = 2 n = 3 n = 4 n = 520 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E n of the (19) for different iteration numbers k and λ = √ ω = 1 and ℓ = 1 / . FIG. 2: The wavefunctions of the two electron interacting in the harmonic oscillator potential field.The parameters λ = √ ω = 1 . V. ACKNOWLEDGEMENT
The research was supported by the Scientific and Technological Research Council ofTURKEY (T ¨UB˙ITAK).
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