The eigenvalue problems for Schrödinger equation and applications of Newton s method. Abstract

Transcription

1 The eigenvalue problems for Schrödinger equation and applications of Newton s method Kiyoto Hira Sumiyoshi, Hatsukaichi, Hiroshima , Japan (Dated: December 31, 2016) Abstract The Schrödinger equation in Woods-Saxon potential for S-state was considered. The one solution was obtained analytically from the Ishidzu s theory 1 stated in this paper. The another one was obtained numerically using the Runge-Kutta method. In obtaining these two solutions the Newton s method played key roles. Finally we compared these two results. And we see that our results are very compatible each others. 1

2 I. INTRODUCTION Schrödinger equations with some types of potentials are possible to solve analytically. And the attempts to solve them analytically have been reported in the literatures until recently. But as is well known, we have a small number of the eigenvalue problems and scattering ones which can be solved analytically. To my knowledge, three bound state solutions for Schrödinger equation in Woods Saxon potential have been reported in analytical forms only for S-state. The first is the solution by T. Ishidzu 1, the second is one by S. Flügge 2,and the third is one by C. Berkmir et al. 3 We are very familiar with the Woods Saxon potential in Nuclear Physics and these three solutions are also very interesting from the point of view of mathematical physics. The Ishidzu s solution and the Flügge s one might be essentially equal in content. Among those three solutions, we refer to the works of T. Ishidzu and S. Flügge. However we do not pursue the work of C. Berkdemir et al. That is why the authors made errors in application of the Nikiforov-Uvarov method 4, which led to incorrect results 5. Ishidzu and Flügge presented in their papers the transcendental equations the roots of which determine the energy eigenvalues. However they did not show us how to solve them explicitly. Although this eigenvalue problem has not been addressed in so many works, our presentation is new in some points. the first point is to demonstrate how to solve the transcendental equation explicitly using Newton s method. the second point point is to demonstrate that the solution obtained from the transcendental equation is fully compatible with the numerical solution obtained in different way using the Runge-Kutta method. The last point is to demonstrate clearly that the Newton s method plays key roles in obtaining these two solutions. As our treatment stated below is transparent to understand logically, we believe that most undergraduates beginning to learn quantum mechanics could follow our approaches easily and they may be pedagogical for those students. And also one of motivations is to demonstrate that the Newton s method is a very useful tool for the eigenvalue problems. 2

3 II. STARTUP. The radial Schödinger equation u(r) of a neutron in the symmetrical Woods-Saxon potential V (r) is given by d 2 u(e, r) d 2 r + 2m l(l + 1)ħ2 (E V (r) )u(e, r) = 0, (1) ħ2 2mr 2 where total wave function for a neutron is represented by ψ(r) = r 1 u(r)y lm (θ, ϕ), and V (r) is given by V (r) =. 1 + e r R a In this paper we take the parameters in Woods Saxon potential as a = 0.67fm, V 0 r 0 = 1.27fm, V 0 = ( (N Z) A )Mev, R = r 0A 1/3 fm, A:mass number, Z:atomic number, N:number of neutrons, A = 208, Z = 82. These parameters are adopted from Blomqvist and Wahlborn(1960) 6. And also the mass of neutron mc 2 = 940Mev and ħc = 197.3Mev fm are taken. We note that for l 0, the equation (1) will be easy to solve numerically but almost impossible to solve analytically. However as Ishidzu calculate, only in the case l = 0, we can solve the equation analytically. So, in this paper we restrict our considerations to the S-state, l = 0. We solve the equation numerically in the case of E < 0, the bound states, and on the boundary conditions u(e, r) = 0 at r = 0, and u(e, r) 0 for r. (3) With these conditions we can determine the energy eigenvalues numerically. (2) III. TO SOLVE A SECOND-ORDER DIFFERENTIAL EQUATION USING RUNGE-KUTTA METHOD. To solve the equation (1) numerically we rewrite it to d 2 u(e, r) d 2 r = g(r)u(e, r), (4) 3

4 where g(r) = 2m ħ (E V 0 ). (5) e r R a If we put v(r) = du(r)/dr, the equation(4) is rewritten to a system of first-order differential equations as follows: du = v, dr (6) dv = g(r)u. dr (7) We apply the fourth-order Runge-Kutta method 7 for the integration of our differential equations. Let the interval of integration as [a, b] and divide it into N intervals of h each such that h = (b a)/n. Then we put r 0 = a, r 1 = r 0 + h,, r N = b and set u i = u(r i ) and v i = v(r i ). If we solve recurrence relation numerically with the initial values u 0 = u(r 0 ) and v 0 = u (r 0 ), we can obtain every {u i, v i }(i = 0, 1,, N). IV. TO FIND THE ROOT OF f(x)=0 NUMERICALLY USING THE NEWTON S METHOD 9. Now we seek the root x of a differentiable function f(x) = 0 numerically using the Newton s method. Suppose x 0 is an approximate root of f(x) = 0 and let δx 0 be the correction to x 0 such that f(x 0 + δx 0 ) = 0, we expand f(x 0 + δx 0 ) in the power series of δx 0. Keeping terms only to first order of δx 0, Then, we have accordingly we obtain f(x 0 + δx 0 ) = f(x 0 ) + δx 0 f (x 0 ) +. (8) f(x 0 + δx 0 ) = f(x 0 ) + δx 0 f (x 0 ) = 0. (9) δx 0 = f(x 0) f (x 0 ), (10) x 1 = x 0 + δx 0 = x 0 f(x 0) f (x 0 ), (11) if we replace x 0 and x 1 for x n and x n+1 respectively,we have the recurrence relation x n+1 = x n f(x n) f (x n ) 4 (n = 0, 1, ). (12)

5 Iterating this recurrence relation, the sequence {x n } is expected to converge to an exact root. When we use this relation, the choice of starting value x 0 is important. If f(x) has analytic expression, we can calculate f (x) analytically. Usually we can compute it numerically using the following relation: f (x n ) = f(x n + h) f(x n h), (13) 2h where h is taken sufficiently small, and f(x + h) and f(x h) are numerically computed. V. ISHIDZU S APPROXIMATE ANALYTICAL EXPRESSION FOR S-STATE IN THE WOODS-SAXON POTENTIAL WHICH DETERMINES THE ENERGY EIGENVALUES. In this section we summarize the Ishidzu s theory on this matter along the lines of his paper 1. Defining r = Rρ, a = αr, V 0 = v 2 0ħ 2 /2mR 2, E = κ 2 ħ 2 /2mR 2, (14) and furthermore changing the variables by u(r) = e κρ χ(x), x = exp {(1 ρ)/α} = exp{(r r)/a}. (15) The differential equation (1) becomes in the case of l = 0 χ (x) κα χ (x) + α2 v0 2 x x(x 1) χ(x) = 0, ( e1/α < x < 0), (16) which can be solved by means of the hypergeometric function 7. For the solution χ(x) for this equation to take a finite value for x = 0 (r ), χ(x) must be χ(x) = F (µ, µ; 1 + 2κα x), (17) where F is the hypergeometric function, and we put µ = α(κ + iκ ), µ = α(κ iκ ), κ = v 2 0 κ 2. (18) The another boundary condition u(r) = e κρ χ(x) = 0 at r = 0 (x = e 1/α ) yields F (µ, µ; 1 + 2κα e 1/α ) = 0. (19) 5

6 Since generally e 1/α 1, now with the help of the relation between hypergeometric functions: Γ(c)Γ(b a) F (a, b; c z) = Γ(b)Γ(c a) ( z) a F (a, a c + 1; a b + 1 1/z) Γ(c)Γ(a b) + Γ(a)Γ(c b) ( z) b F (b, b c + 1; b a + 1 1/z), (20) the equation (19) is reduced to F (µ, µ; 1 + 2κα e 1/α ) Γ(1 + 2κα)Γ( µ µ) = e µ/α F (µ, µ 2κα; µ µ + 1 e 1/α ) Γ( µ)γ( µ + 1) Γ(1 + 2κα)Γ(µ µ) + e µ/α F ( µ, µ 2κα; µ µ + 1 e 1/α ) = 0. (21) Γ(µ)Γ(µ + 1) Now from the fact that a = 0.67fm,R = 7.52fm, e 1/α = exp( 7.52/0.67) = 1.34E 5, the hypergeometric function F in the right-side of the equation(21) could be replaced with unity in fairy good approximation keeping only the first term in the power series of ( e 1/α ) of the function F. Consequently the equation (21) is approximated to e κ Γ(1 + 2κα){ Γ( 2iκ α) Γ( µ)γ( µ + 1) e iκ + complex conjugate} = 0. (22) If we set θ = arg Γ( 2iκ α) Γ( µ)γ( µ + 1), (23) The equation (22) gives us useful relations cos(θ κ ) = 0, κ θ = (n 1 )π (n = 1, 2, ). (24) 2 With the help of the equation (23),the equations (24) become κ + argγ(2iκ α) + argγ( µ) + argγ( µ + 1) = (n 1 )π (n = 1, 2, ). (25) 2 Considering the fact κ, κ, and µ are depending on E from(14) and (18), the roots E n of these transcendental equations corresponding to each positive integer n (n = 1, 2, ) are energy eigenvalues. 6

7 VI. NUMERICAL CALCULATIONS AND CONSIDERATIONS First, we consider the Ishidzu s approximate analytical expression(25) and rewrite it, thanks to mathematical properties of Gamma function 7, to κ + argγ(1 + 2iκ α) 2argΓ(µ) arg(µ) = nπ (n = 1, 2, ). (26) The solutions for these equations give the energy eigenvalues E n (n = 1, 2, ). Now we decompose the equations (26) into the two functions arg(e) and h(e, n) (n = 1, 2, ) such that arg(e) = κ + argγ(1 + 2iκ α) 2argΓ(µ) arg(µ), (27) h(e, n) = nπ (n = 1, 2, ). (28) Now we recognize that the expression (27) is not yet well suited for our practical calculations. For that we rewrite it to a form easy to handle. From the equation 8 Γ(x + iy) Γ(x) where γ is Euler s constant, = e iγy x(x + iy) 1 1 [ 1 + iy/(x + m) ]eiy/m, (29) m=1 the argument of Γ(x + iy), denoted by θ(x, y), is easily given by θ(x, y) = γy tan 1 ( y x ) + [ y m y tan 1 ( )]. (30) x + m m=1 Thanks to this equation, equation (27) is rewritten to a tractable form arg(e) = κ [tan 1 ( 2ακ m ) 2 ακ tan 1 ( ακ + m )] + tan 1 ( κ ). (31) κ m=1 Keep in mind that in our practical calculations the summation in equation (31) is carried out from m=1 to m= Furthermore when α goes to zero in the equation(31), the equation(26) becomes κ + tan 1 ( κ ) = nπ (n = 1, 2, ), (32) κ which is the celebrated eigenvalue condition of square potential well. Now we can plot arg(e) and h(e, n) (n = 1, 2, ) against E as in the Fig.1 and we can see the intersection points satisfy the equation (26) and give the energy eigenvalues. When 7

8 we want to obtain the exact eigenvalue numerically, we need only to apply the Newton s method to the equation f(e) = 0 where f(e) is defined by f(e) = arg(e) h(e, n). From Fig.1 we see the energies E n of the intersections for n = 1,2,3, and 4 are approximately equal to 40.0, 29.0, 15.0, and 1.0 Mev s respectively arg n=1 n=2 n=3 n=4 10 [radian] E[Mev] FIG. 1. The rising curve with respect to E represents arg(e) and horizontal lines correspond to h(e, n) for each integer n = 1, 2, 3, and 4. Then, using Runge-Kutta method, we integrate the equations (6) and (7) over the interval [a, b](a < b) where a is very small and b is a sufficiently greater than R (Nuclear radius). This time, at the point r = c between r = a and r = b so that a < c < b, we split the interval [a, b] into the two intervals [a, c] and [c, b]. We integrate the equations over each intervals using Runge-Kutta method. For the interval [a, c] we integrate forward from r = a to r = c with the initial conditions χ(e, a) a l+1 and χ (E, a) (l + 1)a l (a 0). For the interval [c, b] we integrate backward from r = b to r = c with the initial conditions 2m χ(e, b) exp( ( E) b) and χ 2m 2m (E, b) ( E)exp( ( E) b). Then let ħ 2 ħ 2 ħ 2 denote the wave function at r = c integrated from the inner as χ(e, c ) and let denote the wave function at r = c integrated from the outer as χ(e, c+), and when we write the differrence between their Logarithmic derivatives as f(e, c), then it is given by f(e, c) = χ (E, c+) χ(e, c+) χ (E, c ) χ(e, c ). (33) 8

9 Method Ishidzu formula Runge-Kutta Approximate energy(mev) Convergence value(mev) Approximate energy(mev) Convergence value(mev) Approximate energy(mev) Convergence value(mev) Approximate energy(mev) Convergence value(mev) TABLE I. The comparisons between Ishidzu formula and Runge-Kutta method. In Runge Kutta method we conveniently adopt the parameters explained in this section as a = fm, b = 25.0fm, c = 6.0fm and N = 500. Each approximate energys are estimated from Fig.1. Now we can obtain the energy eigenvalue if we can determine the roots of f(e, c) = 0, the continuity of the Logarithmic derivative of the wave function at r = c. That is a easy task for the Newton s method. Finally, we present the Table.1 to compare with the results calculated for each individual case using Ishizdu s approximate analytical expression and also using Runge-Kutta method. From these results we can emphasize that the results in Table.1 obtained by these two different methods are in fully good agreement each other. Understandably we use the Newton s method jointly together in order to obtain the converge values together with those methods. In conclusion our calculations will leads to the following fact: Although Ishidzu analytic approximate expression explained in this paper is little known in the books on quantum mechanics in contrast to the eigenvalue problem for the 3-dimensional Square well potential, It is stressed that Ishidzu Analytic approximate Expression has a remarkable accuracy and is very interesting from the mathematical point of view. And our approaches may be of interest to those instructors wishing to introduce the practices for eigenvalue problems into their courses. 9

10 permanent address: Sumiyoshi, Hatsukaichi, Hiroshima , Japan 1 T.Ishidzu 1968 Analytical Solution for S-State with the Woods-Saxon Potentials Prog. Theor. Phys S. Flügge 1974 Practical Quantum Mechanics, Springer VerLag, New York Heidelberg Berlin 3 C. Berkdemir,A. Berkdemir,and R.Server 2005 Polynomial solutions of the Schrodinger equation for the generalized Woods-Saxon potential Phys.Rev. C A. F. Nikiforov and V. B. Uvarov 1988 Special Functions of Ma thematical Physics,Birkh äuser,basel 5 C. Berkdemir,A. Berkdemir,and R.Server 2006 Editorial Note: Polynomial solutions of the Schrdinger equation for the generalized Woods-Saxon potential Phys. Rev. C Blomqvist,J. and Wahlborn,S Shell Model Calculations in the Lead Region with a Diffuse Nuclear Potential Airkiv Fysik M. Abramowitz and I. A. Stegun 1972 Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York 8 A. Erdelyi,W.Magnus,F.oberhettinger, and F. G. Tricomi 1953 Higher Transcendental Functions,3 volumes, McGraw-Hill Book Co., New York 9 Atarashi,M. Hira,K. and Narumi,H On the Kaon-Nucleus Optical Potential at Low Energy Prog. Theor. Phys