Dear editor, Thank you for giving me your precious time. I am sorry to bother you again and again.

Transcription

1 Dear editor, Thank you for giving me your precious time. I am sorry to bother you again and again. I will send the editor a revised manuscript which takes into account suggestions from the journal attached below. In accord with suggestions from the editor, I have corrected the language errors he had suggested and in addition I have amended some explanations below equation (22) in the manuscript in order to remove the ambiguities relating to equation (23). Yours sincerely, Kiyoto Hira The comments from the mail on 2018/07/ Referee Comments: Dear colleague, The referee has approved your changes. However you have not incorporated most of the language changes we had suggested. In addition could you provide a derivation of Eq. 23? Also, if -exp(-1/alpha) >>1 then how can one keep only the first term in the Taylor expansion, which is 1? Once the above issues are resolved we can consider your article for publication. Yours sincerely, Editor,

2 Newton s method and energy eigenvalue problems for the Schrödinger equation Kiyoto Hira Sumiyoshi, Hatsukaichi, Hiroshima , Japan da43827@pb4.so-net.ne.jp Submitted on Abstract For a pedagogical example, we take up Newton s method and energy eigenvalue problems for the Schrödinger equation. Newton s method is systematically used to obtain energy eigenvalues and energy eigenfunctions of the Schrödinger equation. The Schrödinger equation with the Woods-Saxon potential is considered for an S-state. One solution is obtained analytically by means of the hypergeometric series. Another one is obtained numerically using the Runge- Kutta method. Because our approach makes the most of Newton s method in this paper, our calculations would have pedagogical benefits for those undergraduate students beginning to learn practical computations actively in physics. 1 Introduction As is well known, Newton s method is very celebrated in mathematical analysis. However, it is less well-known in practical learnings of physics and chemistry. Therefore attempts to apply it to some topics of them seem to be very interesting. In order to use Newton s method for exercises of Quantum mechanics, we take up an eigenvalue problem for the Schrödinger equation with the Woods-Saxon potential, which is very popular in the textbooks of nuclear physics, as a pedagogical example. We use two distict approaches to deal with this problem. One solution was obtained analytically with the help of the hypergeometric series by T. Ishidzu [5]. Another one is obtained numerically using the fourth-order Runge- Kutta method [1]. We would like to stress that in both approaches the Newton s method plays key roles. Motivations of this paper consist of four viewpoints: to show how to solve a transcendental equation using Newton s method without recourse to the graphical method, to show that the solution to the transcendental equation is fully compatible with the numerical one obtained by the distinct method using the Runge-Kutta method, to explain explicitly that Newton s method plays key roles in obtaining these two solutions, and to show clearly, thanks to the transcendental equation, that when the diffuseness parameter of the Woods-Saxon potential gets closer and closer to zero, energy eigenvalues, as expected, converge to those of the 3-dimensional square-well potential. Tools for our approaches are explained briefly in the following four sections after the introduction and section 6 is devoted to numerical calculations and results. Newton s method explained in this paper has been applied to the complex energy eigenvalues problems of kaonic atoms for the first time in the work of M. Atarashi et al. [2] and the present paper originated from their work. Volume/Issue/Article Number 1

3 2 Startup The radial Schödinger equation u(r) of a neutron in a symmetrical Woods-Saxon potential V(r) is given by d 2 u(e, r) dr 2 + 2m l(l + 1) h2 (E V(r) 2 h 2mr 2 )u(e, r) = 0, (1) where the total wave function ψ(r) for the neutron is represented by ψ(r) = r 1 u(r)y l,m (θ, φ), and V(r) is given by V(r) = V 0. (2) 1 + e r R d In this paper we take the parameters in the Woods- Saxon potential for convenience as A = 208, Z = 82, d = 0.67 fm, r 0 = 1.27 fm, V 0 = ( (N Z) A ) Mev, R = r 0A 1/3 fm. These parameters are adopted from Blomqvist and Wahlborn [3] and give us V 0 = MeV, R = fm. Besides, the mass of neutron mc 2 = 940 Mev and hc = MeV fm are taken. We note that for l = 0, equation (1) is easy to solve numerically but impossible to solve analytically. However, as Ishidzu calculated, only for l = 0 we solve the equation analytically. For this reason, in this paper, we restrict our considerations to an S-state (l = 0). When E < 0 we solve equation (1) numerically with condition that (3) u(e, r) = 0 at r = 0 and u(e, r) 0 for r. (4) With these boundary conditions we can determine the energy eigenvalues numerically. 3 Runge-Kutta method In order to solve equation (1) numerically in case l = 0 we rewrite it as d 2 u(e, r) dr 2 = g(r)u(e, r), (5) where g(r) = 2m h 2 (E V 0 ). (6) 1 + e r R d If we put v(r) = du(r)/dr, equation (5) is changed to a couple of the first-order differential equations as follows: du = v, dr (7) dv = g(r)u. dr (8) We apply the fourth-order Runge-Kutta method [1] for integration of the differential equations (7) and (8). Let [a, b] stand for an interval of integration and we divide it into N intervals of width 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 the recurrence relations resulting from the Runge-Kutta method 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). 4 Newton s method We find a solution x of a differentiable function f (x) = 0 numerically. Suppose x 0 is an approximate solution to 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 powers of δx 0 as f (x 0 + δx 0 ) = f (x 0 ) + δx 0 f (x 0 ) +. (9) Keeping terms up to the first order in δx 0, f (x 0 + δx 0 ) = f (x 0 ) + δx 0 f (x 0 ) = 0. (10) Then we obtain δx 0 = f (x 0) f (x 0 ). (11) Accordingly we obtain, by letting x 1 = x 0 + δx 0, x 1 = x 0 f (x 0) f (x 0 ). (12) Volume/Issue/Article Number 2

4 If we replace x 0 and x 1 with x n and x n+1 respectively, we obtain the recurrence relations x n+1 = x n f (x n) f (x n ) (n = 0, 1, ). (13) By iterating these recurrence relations, the sequence {x n } is expected to converge to an exact solution. When we use these relations, a choice of the starting value x 0 is important. If f (x) has an analytic expression, we can calculate the derivative f (x) analytically. Usually, we can easily compute the derivatives numerically using the following relations: f (x n ) = f (x n + h n ) f (x n h n ) 2h n (n = 0, 1, ), (14) where h n are taken sufficiently small, and f (x n + h n ) and f (x n h n ) are numerically computed. 5 Ishidzu s analytical solution for S-state with the Woods-Saxon potential In this section we explain briefly Ishidzu s theory along the lines of his paper [5]. Putting r = Rρ, d = αr, V 0 = v 2 0 h2 /2mR 2, E = κ 2 h 2 /2mR 2, and furthermore changing the variables by x = exp {(1 ρ)/α} = exp{(r r)/d}, u(r) = e κρ χ(x), equation (1) becomes in case of l = 0 χ (x) κα χ (x) + α2 v 2 0 x x(x 1) χ(x) = 0 (15) (16) ( e 1/α < x < 0), (17) which can be solved by means of the hypergeometric series [1]. For the solution χ(x) to this equation to take a finite value at x = 0 (r ), χ(x) must be χ(x) = F(µ, µ; 1 + 2κα x), (18) where F is the hypergeometric series [1], and we put µ = α(κ + iκ ), µ = α(κ iκ ), κ = v 2 0 κ2. (19) The other 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. (20) Since generally e 1/α 1, the function F must be continued analytically. By use of the relation between the hypergeometric series [1]: F(a, b; c z) Γ(c)Γ(b a) = 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), (21) where Γ is the Gamma function, equation (20) is reduced to F(µ, µ; 1 + 2κα e 1/α ) = Γ(1 + 2κα)Γ( µ µ) Γ( µ)γ( µ + 1) e µ/α F(µ, µ 2κα; µ µ + 1 e 1/α ) + Γ(1 + 2κα)Γ(µ µ) Γ(µ)Γ(µ + 1) e µ/α F( µ, µ 2κα; µ µ + 1 e 1/α ) = 0. (22) Since d = 0.67 fm, R = 7.52 fm, and e 1/α = 1.34E 5 1, we keep only the first term of the expansion in powers of ( e 1/α ) of the hypergeometric series F. Therfore, the function F on the right-side of equation (22) can be approximated as unity to fairly good approximation. Consequently equation (22) becomes e κ Γ(1 + 2κα) { Γ( 2iκ α) Γ( µ)γ( µ + 1) e iκ + Γ(2iκ α) Γ(µ)Γ(µ + 1) eiκ } = 0, (23) where we have used equation (19) and have replaced the function F on the right-side of equation (22) with unity. If we set θ = arg Γ( 2iκ α) Γ( µ)γ( µ + 1), (24) Volume/Issue/Article Number 3

5 where the notation arg represents the argment of a complex number, equation (23) gives us significant relations cos(θ κ ) = 0, κ θ = (n 1 )π (n = 0, ±1, ±2, ). (25) 2 With the help of equation (24), equations (25) become κ + argγ(2iκ α) + argγ( µ) + argγ( µ + 1) = (n 1 )π (n = 0, ±1, ±2, ). (26) 2 As κ, κ, and µ depend on E through equations (15) and (19), the solutions E n to these transcendental equations (26) are energy eigenvalues. 6 Numerical calculations and the results We consider Ishidzu s approximate analytical expressions (26) and rewrite them, thanks to the mathematical properties of the Gamma function [1], as κ + argγ(1 + 2iκ α) 2argΓ(µ) arg(µ) = nπ (n = 0, ±1, ±2, ). (27) we decompose equations (27) into two functions arg(e) and h(n) (n = 0, ±1, ±2, ) such that arg(e) = κ + argγ(1 + 2iκ α) 2argΓ(µ) arg(µ), (28) h(n) = nπ (n = 0, ±1, ±2, ). (29) We recognize that expression (28) is not yet well suited for our straightforward calculations. For that we deform it to a distinct form that is easy to calculate. By use of the following equation [4]: Γ(x + iy) Γ(x) = e iγy x(x + iy) 1 1 [ 1 + iy/(x + m) ]eiy/m, m=1 (30) where γ is the Euler s constant, the argument of Γ(x + iy) is easily given by argγ(x + iy) = γy tan 1 ( y x ) + m=1[ y m y tan 1 ( x + m )]. (31) Thanks to this equation, equation (28) can be deformed to a tractable form arg(e) = κ + tan 1 ( κ κ ) m=1 [tan 1 ( 2ακ m ) 2 ακ tan 1 ( ακ + m )]. (32) Keep in mind that for practical calculations of the values of the function arg(e), equation (32) is used and its summation is carried out from m = 1 to m = When α goes to zero in equation (32), equation (27) becomes κ + tan 1 ( κ ) = nπ (n = 0, ±1, ±2, ), (33) κ which is changed to κ tan κ = κ. (34) This is the celebrated eigenvalue condition [6] of the 3-dimensional square-well potential (SQWP). From this it is shown explicitly that Ishidzu s expression is a mathematical extension from the 3-dimensional square well potential to the Woods-Saxon potential. Now we can plot arg(e) and h(n) (n = 1, 2, ) against E as in Figure 1, which shows us that the intersection points satisfy equations (27) and give the energy eigenvalues. When we want to obtain the exact eigenvalues numerically, we need only to apply Newton s methods to the equation f (E, n) = 0, where f (E, n) is defined by f (E, n) = arg(e) h(n). From Figure 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 respectively. These values can be adopted as the starting values for Newton s method. Volume/Issue/Article Number 4

6 [radian] E[Mev] arg n = 1 n = 2 n = 3 n = 4 Figure 1: The increasing curve with respect to E represents arg(e) and horizontal lines correspond to h(n) for each integer n = 1, 2, 3, and 4. In what follows, using the Runge-Kutta method we integrate equations (7) and (8) over an interval [a, b](a < b), where a is very small and b is sufficiently greater than R (Nuclear radius). This time, at an intermediate 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 interval using the 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, l = 0). For the interval [c, b] we integrate backward from r = b to r = c with the initial conditions χ(e, b) exp( 2m ( E) b) and χ (E, b) h 2 2m 2m Method Ishidzu Runge-Kutta Approximate energy (MeV) Convergence value(mev) SquareWell (MeV) Approximate energy (MeV) Convergence value(mev) SquareWell (MeV) Approximate energy (Mev) Convergence value(mev) SquareWell (MeV) Approximate energy (MeV) Convergence value(mev) SquareWell (MeV) no-solution no-solution ( E)exp( ( E) b). Then let χ(e, c ) denote the wave function at r = c integrated from r = a h 2 h 2 Table 1: The comparisons between Ishidzu formula and the Runge-Kutta (R-K) method. For the R-K to r = c and let χ(e, c+) denote the wave function at method we adopt the parameters of the intervals explained in this section as a = 0.0 fm, b = 25.0 fm, r = c integrated from r = b to r = c, and if we write the difference between their logarithmic derivatives as c = 6.0 fm, and N = 500 for convenience. Each f (E, c), then it is given by of the approximate energies are estimated from FIG. f (E, c) = χ (E, c+) χ(e, c+) χ (E, c ) χ(e, c ). (35) 1. The eigenvalues for the 3-dimensional square-well potential are calculated using equation (33) and also Now we can obtain an energy eigenvalue if we calculated from the R-K method using equation (35). can determine a solution E to f (E, c) = 0, the continuity When the R-K method is applied, to avoid the singu- of the logarithmic derivative of the wave function larity of the square-well potential at r = R, the follow- at r = c. This is an easy task for Newton s method. ing recipes are taken: The endpoint c of the inner interval We need to verify that the solution E of f (E, c) = 0 is [0, c] is shifted from c = R to c = R free of c. ; the endpoint c of the outer interval [c, b] is shifted from c = R to c = R Volume/Issue/Article Number 5

7 r E 1 (MeV) E 2 (MeV) SQWP r E 3 (MeV) E 4 (MeV) no-solution no-solution no-solution no-solution SQWP no-solution Table 2: The energy eigenvalue dependence on the diffuseness parameter d. r is defined as the diffuseness parameter d divided by 0.67 fm (d = r 0.67 fm). The subscript numbers of E correspond to the number n in equation (27). E n (n = 1, 2, 3, 4) are the calcuated values for r using equation (27). 7 Concluding remarks We have demonstrated that Newton s method is a clearly powerful technique for solving eigenvalue problems of quantum mechanics and also have endorsed the validity of Ishizu s analytical solution numerically using the Runge-Kutta method. Although most of books on quantum mechanics are unfamiliar with Ishidzu analytic approximate expression explained in this paper in contrast to the eigenvalue problem for the 3-dimensional square-well potential, it should be noted that Ishidzu analytic approximate expression has a good accuracy and is very interesting from the viewpoint of mathematical physics. In particular, we believe that most novice students beginning to learn practical uses of quantum mechanics could follow our calculations easily, which may have pedagogical merits for those students. Our approaches may be of interest to those instructors who would like to introduce applications of Newton s method to various fields positively into their courses. Finally, we present Table 1 to compare the results calculated using Ishizdu s approximate analytical expression with those calculated using the Runge-Kutta method. From these two results we can emphasize that the results obtained by these two distinct methods are in good agreement with each other. Understandably we use Newton s method in order to obtain the convergence value in each individual method. Furthermore we calculate the energy eigenvalues of the 3-dimensional square-well potential with the same V 0 and R in equation (3) for the S-state (l = 0) using Newton s method. We include them in Table 1 for comparison. Besides, by using Ishidzu s expression we demonstrate explicitly in Table 2 that when the diffuseness parameter d of the Woods-Saxon potential approaches zero, the energy eigenvalues, as expected, converge to those of the 3-dimensional square-well potential. Acknowledgments I would like to thank the editor for reading my manuscript carefully, pointing errors and ambiguities, and giving me fruitful comments. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables Dover Publications, Inc, New York, [2] M. Atarashi, K. Hira, and H. Narumi, On the kaon-nucleus optical potential at low energy, Prog. Theor. Phys, 60, , [3] J. Blomqvist and S.Wahlborn, Shell model calculations in the lead region with a diffuse nuclear potential, Airkiv Fysik, 16, 545, Volume/Issue/Article Number 6

8 [4] A. Erdelyi, W. Magnus, F. oberhettinger, and F. G.Tricomi, Higher Transcendental Functions volume 3, McGraw-Hill Book Co., New York, [5] T. Ishidzu, Analytical solution for S-state with the Woods-Saxon potentials, Prog. Theor. Phys, 40, , [6] L. Schiff, Quantum Mechanics 3rd. ed., Mc- GrawHill, New York, Volume/Issue/Article Number 7

9 avqj > a[r,r(z] Newton's method and energy eigenvalue problems for the Schriidinger equation Kiyoto Hiral 1 SumiyoshlHatsukaich,Hiroshima , Japan da43827@pb4. so-net. ne. jp Submitted on Abstract For a pedagogical example, we take up Newton's method and energy eigenvalue problems for the Schrcidinger equation. Newton's method is systematically used to obtain energy eigenvalues and energy eigenfunctions of the Schrcidinger eq uation. The Schrcidinger equation with the Woods-Saxon potential is considered for an S-state. One solution is obtained analytically by means of the hypergeometric function. Another one is obtained numerically using the Runge-Kutta method. Because our approach makes the most of Newton's method in this paper, our calculations would have pedagogical benefits for those undergraduate students beginning to learn practical computations actively in ohvsics.. ' ' trfic0rrcrt'sfqc!$t 1!ntroduction \,, by T. Ishidzu[S]. Another one is obtained numeqically using the fohrth-order Runge-Kutta methodfj. We would like to stress that in both approaches l$ewton's method plays key roles. Motivations of this paper consist of four viewpoints: to show how to solve a transcendental equation using th,enewton's method without recourse to -l\qgraphical method, to show that the solution to the transcendental equation is fully compatible with the iln".ffi:.?lt'::ffi jffi,tfl'iln::l"fl ilf \\ Newton's method plays key roles in obtaining two solutions, and to show clearly, thanks to the transcendental equation, that when the diffuseness parameter of the Woods-Saxon potential gets closer and \\ \\ closer to zerot energy eigenvalues, as expected, con- I \ verge{ to those of the 3-dimensional sqllare well po- \ \ tential. t \ As is well known, Newton's method is very celebrated in mathematical analysis. Howeveg it is lit- Tools for our approaches are explained briefly in tle known in practical learnings of physics and chem- the following four sections after the introduction and istry. Therefore attempts to apply it to some topics -15u,rectron 6 is devoted to numerical calculations and of them seem to be very interesting. In order to use results. Newton's method for exercises of physics, we take up an eigenvalue problem for the Schrcidinger equation with the Woods-Saxon potential, which is very popular in the texlbooks of nuclear physics, as a pedagogical example. We use two different approaches to deal with this problem. One solution is dbtained analytically with the help of the hypergeometric fnncfion ThrNewton's method explained in this paper has been applied to complex energy eigenvalues problems of kaonic atoms for the first time in the work of M. Atarashi et al.[2] and the present paper originated from their work. Volume/Issue /Article Number

10 \\ 2 Startup The radial Schodinger equation tt(r) ola neutron lna symmetrical Woods-Saxon potential 1/(r) is given by dzu(e,r) d.r' +24e, - vu)- /(/ + 1)fi2 ) rt(e,r) : Q, (1) h2'- ' \'/ 2mr2 where the total wave function r/(r) for the neutron is and tffe V (r) represented by QU) : r-1u(r)y1,,,,(0, fi, is given by v(r): '0,'-,.. t t e--t' In this paper we take the Parameters in the Woods- Saxon potential for convenience as A:208, d = 0.67fm, vs : ( ;?)Mev, r.j These parameters are adopted Wahlborn[3] and give us vo : - M.0L Mev, R : frn. And also the mass of neutron mcz Mev andhc : 197.3Mev 'fm are taken. r,j fre note that for I f O, eqtation (1) will be easy. to solve numerically but impossible to solve analyti- \ \ cally. Howevergas Ishidzu calculated, only for I : 0 \ \ the equation analytically' So, in this paperq\,ve restrict our considerations to an S-state (l : 0)' W6" f ( 0 we solve equation (1) numerically oylthe boundary conditions that wt\/\ tl(e,r):oatr:o andu(e,r) +0 forr-+oo' (4) With these conditions we can determine the energy eigenv altres nr'rmerically. (2) Z :82. ro:1.27ff, (3) R: roalftrl., from Blomqvist and where S?) : -Tr'- jb) If we ptrt o(r) : du(r) / dr,equation(5) is changed to a couple of first-order differential equations as follows: [#:. ldo I 4, : *\')"' We apply the fourth-order Runge-Kutta method[1] for integration of the differential equations (7) and (8). Let la,bl stand for an interval of integration and we divide it into N intervals of width h each such that h : (b - a) /N. Then we Plut ro : a, ft : ro+h,"', tn : b and set ui : u(r) artd tti : o(t). If we solve the recurrence relations resulting from the Runge-Kutta method numerically with the initial values uo: u(ro) and o9 : u'(to), we can obtain every {ui,a;}(i : O,\,"',N). 4 Newton's method Now we find a solution * fi**nennable ftinction (*) :0 numerically. Suppose xs is an approximate f soltttion to f (*) :0 and let drs be the correction to rs such that /iro t dxo) : 0.W expand /(rs * dxs) in a power series of dxs' f (*o + 6xs) : /(ro) + 6xnf' (x,s) + " ' Keeping terms up to the first ordurfr 6*0, f (*o + 6xs) : f (xil + Sxsf' (xr1 : s. Then we obtain (6) (7) (8) (e) (10) \ It 3 Runge-Kutta method In order to solve equation (1) numerically in case I : 0 we rewrite iv "A : g(r)u(e,r), Volume,/ Issue /Article Number (5) (11) Accordingly we obtailby letting xl : xo + 6xo, f xl:xo-t@) (xo) (12) I I

11 \ If we replace.x9 and 11 with x,, and r,1+1 respectively, we obtain the recurrence relations ^11+l-*ll - _ f(r,) t?t/ ' \ J \xtt) Iterating these recurrence relatiors, the sequence {x,, } is expected to converge to an exact solution. When we ttse these relations, a choice of the starting value xg is important. tf f (x) has an analytic expression, we can calctrlate the derivativ u f ' (*) analytically. Usually_rwe can easily compute the derivatives numerically using the following relations: ',/ where F is the hypergeomehk Tgrnlll, Ir : aq< + ir<' ), p : a(r - it<' ), r' The other boundary condition a(r) r:0 (x: -stl") yields F(lr,y;t *2x*l - el/o) : o. and we put n:o'r"")' (13) : \E-A' (1s) Since generally - er/a1>> L,y>* with the help of the relation between tl-re hypergeometric functions [ 1 ]f F (s, t; ulz) : s-kpx(y) :0 at (20) ft(x-\ :.f (x', -f h,,) - f (x,, - h,,) (n :0,1....,t J \^n) - 2hu lr (14) f/rr)f/.s - i). + 1i61ffi (-z)-tf(t,t - utt;t - sttll/z), where h,, are taken sufficiently small, and f (*u + lrr) and f (x" - hn) are numerically computed. 5 lshidzu's analytical solution for S-state with the Woods-Saxon potential In this section we explain briefly the Ishidzu's theory along the lines of his paper[s]. Defining 1 &.t -.-J- "df,)+ }}" r: Rp, d: ar,. i,.vn vo:-.aihz/zntrz, E:-t?h2/2mR2, (15) Nhll""r" t., *. P and furthermore changing the variables by \ \ <i \l ---2 ttlr): e '''X\x), x : - exp {Q - p) / o} : -exp{(r - r) /d},' 4I equation (1) becomes in case of / : 0 x"@)+u?x'q)+ ffix{*):o (-it/o <x<o), (16\ (r7) where I is the Gamma ftrncfion, equation (20) duced to F(tr,it;7 t2xal - r'/n) _r(l+zra)r(y- u) r(p)r(p + 1) e'kf (1. -t Zrca) (21)..ra@ rc re-, x e-t'/of(.l.r,lt - 2r<u;1t - lt + tl -,-t/u) -, r(l + 2xa)f(y - y) r(i,)f (t, + l) x e-ri/,f(f,f - 2rca; p - y + 1l x1 - r-t/ - 0. (22) Now from the fact that d : 0.67fm,R : 7.52fm, \ e-t/o : exp(-7.52/0.67) : 1.34E - 5, the hypeage-, \ olnetric frilitior, f jp*re risht-side of equation(22) \ gor*itf fe replaced with l.ruty fl faif Sood approximatiorl keeping only the first term in the power series of (e-t/t) of the function F. Consequently equation (22) becomes %ffi+u^^? which can be solved by means of the hypergeometric furctijn[1]. For the solution X@) to this equation to takeiffie value for x : 0 (r i *), x(x) must be x(x) : F(lt,F;'J' t 2rculx), (18) Vohrme / Issue / Article Number If we set l( -2ir' a\ 0:ar{ilffi, (23) (24)

12 u \\ where the notation a/g represents the argment of a complex number, equation (23) gives us significant relations cos(0 - K') :0, r' - o : 1" _ : 0,il,+2,...). (25) With the help of equation (24), equations (25) become rct + argl(2ircta) + argf (y) + argf (y + t) 1. :(n-1)n (n:0,+7,+2,"'). (26) C."s$g#f,qj4" fact that x, xt,and p are dependhgon E throfigh.(15) and (19), the solutions E,, to these transcendentallquations (26) are energy eigenvalues. 6 Numerical calculations and the results First, we corsider-the-ishidzu's approximate analytical expressions (26) and rg){rite them, thanks to the mathematical properties of'camma function[] tq 6rn rt + argr(l a2ixta) - zargf (y) - arg(y) :n7r (n:0,+l,t2,"'). (27) 1-. Now we decompose equation (27) into two functions arg(e) andh(n) (n :0,!1,+2, ' ' ') such that arg(e) : rct + argl(l * 2irct a) - zargf (y) - argot), (28) h(n) : nv (n : 0, +L, +2, ' ' '). (29) l$r we recognize that expression (28) is not yet well suited for our straiehtfor.ward calculations. For that we deform it to o ui$#53#* ror* 8o$, t?calcr,- late. With the help of the following equation[4]: r(x + iy) r(r) : s'itlt *1* * iy)-' II )eilt / ",,*lr't+iy/(x+m) where 7 is the Euler's "9lt$ ra Gfre?igument 6n rlx lly\. ba s+d+f-sng :fu'r4* is easily given by argf (x + iy) : -"yy - ''u)+frp-tu''-r( v tan-' (it -r il.,lrp -,^, \ r + *!). 'n*'d- Qt) Thanks to this equatiory equation (28) f deformed to a tractable form 'I arg(e):r'+tan-l1l; - f 1tor,-' f'41-2tan-1 (-.,i\, )1.,t:I ' m' 'atclm" (32) Keep in mind that for practical calculations of the valtres of the function arg(e).equation (32) is used an{ summation is caruied /.,t fro#*=ito{",=1oooooi -i6 Furthermore when a goes to zero in equation (32), equation$ (27) -t become..; r,.kt r * tan-' (-) : nn K which is changed to ll t<tanr : -x. (n : 0, *1.,12,...), (33) (34) This is the celebrated eigenvalue condi9on[6] of the- 3-dimensional square well potential (SQWP).' From this it is shown explicitly tha(+he Ishidzu's expression is a mathemati"a e}t"r,fi3*lrrom the 3-dimensional sqrlare well potential to the Woods-Saxon pofential. Now we can plot arg(e) and h(n) (n : 1,2,' ' ') against E as injhsfigure L, which shows tts that the intersection points satisfy equations (27) and give the energy eigenvalues. When we want to obtain the exact eigenvalues numerically, we need only to apply thr Newton's rnethods to the equation / (E,n) :0, where f (E,") is defined by f (E,n) : ary(e) - h(n). tuom il tl I tle Figure 1 lye see the energies E,, of the intersections I J fof. = L,23?and4 are approximately eqttal to -:0.L -29.0,-L5,0, and-1.0 Mev's respectively..aaa4l[lese M. V values can be adopted as the starting values ftot 1ft (30) Newton's methodf Volume /Issue / Article Number

13 U:L.45 Elrevl -15 Figure 1: The increasing curve with respect to E represents arg(e) and horizontal lines correspond to h(n) for each integer n : 1,2,3, and 4. (\? 7 Thery using the Runge-Kutta method we integrate eqtrations (7) and (8) over an interval la,bl(a < b), where a isvery small and b islsrrfficiently greater than R (Nuclear radius). This time, at an intermediate point r : cbetween r : a. arrd r : b so that a < c < b, we split the interval [a,b] into the two intervals [a, c] xd lc,bl. We integrate the equatioru over each interval using the Runge-Kutta method. For the ilrterval [a, c] we integrate {orward from r : a to r : c with the initial conditions X(E,a) N at+1' andxt(e,a) = (l f L)at (a x 0, I : 0). For the interval [c,b] we integrate backward from r : b to r : c with the initial con- b) and xt(r,b) x ditions x@,b) o,*p1- rff;\ - ffl-e)r-pe fficilb). rhenletx(e,c-) denote the wave ftinction at r : c integrated from the innerdmd let X(E, c+) denote the wave ftmction at r : c integrated from the outer, and if we write the differ- "n.jb"t*""r, their,(ogarl)hmic derivatives as f (E,c), then it is givenby '- ",t/c ^r, X, (E,r_) f(e,c):trh-'ffi (3s) Now we can obtain an energy eigenvalue if we can determine a solution to f (E,c) : 0, the continuity of the Logarithmic derivative of the wave function at r : c. This is an easy task for*he Newton's method. In that case we need to examine the solution E to f (E,c): 0 being free of c. -- pffi4 'l Vohlme /Issue/Arficle Number Table 1: The comparisons between tr&e-ishidzulsformula and the Runge-Kutta method. For the ffi* 4 Kutta method we adopt the parameters of the intervals explained in this section as a : 0.0tm&: 25.0fm, c : 6.0fm and N = 500 for convenience. Each of the aooroximate "r","rnfrllbetimated from FIG. 1. The values for the 3-dimensional square well potential are calculated using equation (33) and also calculated using equation(35). *When the Runge-Ktrtta metho$pplied, to.avoid the singularity of the square well potential atl=b)the following recipes are taken: The end point.g of the inner interval [0,c] is,shifted from-c=r t6,,/ -r=w 6ooooor. aa*ft so f l#ndloi"t c of theglo.4 Xr interval [c,b] is shifted from c=r to c=r*

14 L s -3L l L sqwp t E3(Mev) Ea(Mev) t no-solution no-solution no-solution no-solution sqwp no-solution Table 2: The energy eigenvalue dependence on the difftiseness parameter d. r is defined as the difftiseness parameter d divided by 0.67tm, (d : r * 0.67fm). The subl;cript numbers of E correspond to thililintrmber n in@uation (27). En(n : 1.,2,3,4) are the calcuateci. values for r's using equation (27). 7 Concluding remarks We have illushated that.+he Newton's method is a powerful and indispensable technique for solving p/ eigenvalue problems of qrrantum mechanics and also have endorsed the validity of.tte Ishiztls analytical solution numerically using the Runge-Kutta calctilations have led to the following c-e,acl,t-r-trttl Jaef Although dhe Ishidzrd analytic approximate expression explained in this paper is little known in books on quantrlm mechanics in contrast to the eigenvalue problem for the 3-dimensional Square well potential, it should be noted that the Ishidztreanalytic approximate expression has a good accuracy and is very interesting from the viewpoint of mathematical physics. Further, we believe that most novice students beginrring to leam practical rrses of quanhrm,mechanics could follow our approaches easilyy?r,a"1n"y *uy have pedagogical merits for those students"aad a*t'o 3 w'e nfg,ht be of interest to those instructors who would like to introduce applications of the Newton's method to various {ields positively into their courses. 7" Finally, we present'th,etable 1to compare the results calculated using.se-ishizdu's approximate analytical expression with those calculated using the Runge-Kutta method. From these two results we can emphasize that the results obtained by.these two different methods are in good agreem"nt'ikn otner. u.,- derstandably we use.tihenewton's method jointly together in order to obtain the convergence value in each individual method. Furthelnore we calculate the energy eigenval-.,. ues of the 3-dimensional sqrlare well potential with ll the same Vs and R as in (3) for the S-state (/ : 0) using t t -thealewton's method.adud ds include them in fiq References [1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Thbles Dover Publications, Inc, New York,!972. [2] M. Atarashi, K. Hira, and H. Narum[JOn the kaon-ntrcleus optical potential at low energla) Prog. Theor. Phys, 60, , tg78. ) [3] J. Blomqvist and S.Wahlborrg)Shell model calculations in the lead region with a difftlse nuclear potential Airkiv Fysik, 16, 545, z-. # otl-'' Jil.o Table l for comparison. *n*also using tlrrlshidzu's expression we demonstrate explicitly in Table 2 that when the difftiseness parameter d of the Woods-Saxon potential approaches zero, the energy eigenvalues, as expected, converge to those of the 3-dimensional square well potential. [4] A. Erdelyi, W Magnus, F. oberhettinger, and F. G.Tiicomi. Higher tanscendental Ftmctions, volume3, McGraw-Hill Book Co., New York, [5] T. Ishidzu. Analytical solution for s-state with the, woods-saxon potentialsrprog. TGr. Phys, 40, ,1968. / Volume/Issue/Article Number wwwphysedu.in

15 [6] L. Schiff. Quantum Mechanics 3rd. ed. Mc- GrawHill, New York, Volume/Issue /Article Number