Dear Dr. Kiyoto Hira,

Transcription

1 Dear Dr. Kiyoto Hira, Your paper "Newton's method and energy eigenvalue problems for Schrodinger equation" has been examined by one of our referees and the comments are sent to you for your consideration. You may address the queries, if any, of the referee and make a revised submission within 90 days for further processing. Yours Sincerely, Raghavan Rangarajan Editor, Physics Education Manuscript No: PE Title: Newton's method and energy eigenvalue problems for Schrodinger equation Author Name: Kiyoto Hira Affiliation: Sumiyoshi,Hatsukaichi,Hiroshima, ,Japan Author da43827@pb4.so-net.ne.jp Referee Comments: Dear colleague, The referee report for your article is attached. The referee has approved your article for publication with a suggestion that you may wish to incorporate in your article. However before the article can be accepted for publication it needs to be modified for language errors. The corrections to only the title and abstract will also be sent to you. You are requested to check the entire article for such language issues and resubmit a revised manuscript. Yours sincerely, Editor, Physics Education

2 Referee Report form PHYSICS EDUCATION An e-journal devoted to Physics Pedagogy REPORT OF REFEREE Manuscript No [PE ] Original Please note that this is pedagogical, educational journal addressed to physics students and teachers. CHECKLIST: 1. Is the paper of good quality, free from errors, misconceptions or ambiguities, and does it contain sufficient material of interest to physicists, teachers and students to warrant its publication in Physics Education? 2. Is the paper appropriate for this journal? 3. Is the manuscript a clear, concise, reasonably self-contained presentation of the material giving adequate reference to related work? Is the English satisfactory? Please indicate the changes needed. 4. Are the tables and figures clear and relevant and are the captions adequate? Are they either too many or too few? 5. Does the paper make effective use of journal space, or are unnecessary, unimportant or subject to condensation? 6. Is the title appropriate and is the abstract adequate? 7. Further remarks, if any. Referee report In the manuscript, the author computes the energy eigenvalues and eigenfunctionsfor the S-states (l=0) of the three dimensional time-independent Schrodinger equation (TDSE) with Woods-Saxon potential. The potential is a model of the nucleon-nucleon interaction in the nuclear shell-model calculations. An analytical scheme has been followed to obtain a set of transcendental equations from the TDSE. The author, then, uses Newton's method to obtain the solutions of the transcendental equations, and this point has been very well emphasized and explained for the pedagogical purpose. The solutions to the problem are obtained by using two distinct methods, and then the comparison between the obtained results is shown in the Table-1. The first method is analytic, formulated by Ishidzu (ref. [5]), in which the solutions are obtained from a set of transcendental equations (Eq. 26). This method is well described for the Woods-Saxon, and the derivative of the Woods-Saxon potential in the article by Ishidzu et al (1968). Therefore, the Sec.5 of this paper is a rederivation of the Ishidzu's method, but it helps to present the matter in a self-contained way. The solutions of the transcendental equations are, then, obtained using Newton's method. So, this part of the manuscript is of good pedagogical value. The second method uses Runge-Kutta (RK) integration to solve TDSE. In this case, the radial part of the TDSE is solved in two intervals [a,c] and [c,b] with appropriate boundary conditions at a and b. By

3 defining a function which is the difference of the logarithmic derivatives of the obtained wave functions at c, and then equating this function to zero, the energy eigenvalues of the equations are obtained by Newton's method. This secondapproach is also explained in a lucid manner. The manuscript is well suited for publication as an article in Physics Education. However, I have one minor suggestion: what about including the eigenvalues of the square well potential from the RK method in Table.1? If there are some technical difficulties, then, this question can be ignored. In summary, I recommend the publication of the manuscript as an article in Physics Edication. The author choose to revise the manuscript to incorporate the eigenvalues of the square well potential from the RK method in Table.1. This is, however, optional and the manuscript may be published as it is. RECOMMENDATION (Please tick in the brackets) 1. Publish in the present form ( ) 2. Publish with suggested revision ( ) 3. Revision and re-referral requested ( ) 4. Reject (Not suitable for Physics Education) ( )

4 Dear Editor Thank you for your reply to my submission. And I appreciate the referee for his fruitful suggestions. I have amended my manuscript[pe18_01_483] according to the referee s suggestions. List of amendments: 1. Along the referee s suggestion, I have included the eigenvalues of the square well potential from R-K method in Table 1. In its caption I gave some additional explanations for the R-K method. 2. As I modified my manuscript as to my English errors and send it to the editor, I ask you to replace the latest version of my manuscript with the last one if circumstances permit. Your sincerely Kiyoto Hira Sumiyoshi, hatsukaichi, Hiroshima, , Japan da43827@pb4.so-net.ne.jp files sent to editor: manuscript no: PE18_01_483 lettertoeditor.pdf PE18_01_483_Revised_KiyotoHira.tex PE18_01_483_Revised_KiyotoHira.pdf graph.eps widetext.sty

5 kiyotohira 差出人 : kiyotohira <da43827@pb4.so-net.ne.jp> 送信日時 : 2018 年 7 月 2 日月曜日 20:50 宛先 : 'secretary@physedu.in' 件名 : Query about paper re-submission Dear Editor I have not heard the current status of my manuscript[pe_18_01_483] in the review process. Could you let me know when I can expect notice regarding the decision of the editorial board? Thank you for your time and consideration. I am looking forward to hearing from you. Your sincerely, Kiyoto Hira. 1

6 kiyotohira 差出人 : Sandip Wadhai <secretary@physedu.in> 送信日時 : 2018 年 7 月 4 日水曜日 0:45 宛先 : kiyotohira 件名 : Re: Query about paper re-submission Dear Sir, Current status of your manuscript shows "Resubmitted article". Your article is pending with final revision of referee. We will let you know the updated status, once response for your manuscript received from Referee. Regards, Secretary, Physics Education. On Mon, Jul 2, 2018 at 5:19 PM, kiyotohira <da43827@pb4.so-net.ne.jp> wrote: Dear Editor I have not heard the current status of my manuscript[pe_18_01_483] in the review process. Could you let me know when I can expect notice regarding the decision of the editorial board? Thank you for your time and consideration. I am looking forward to hearing from you. Your sincerely, Kiyoto Hira. 1

7 Newton s method and energy eigenvalue problems for the Schrödinger equation Kiyoto Hira 1 1 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 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 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 the Newton s method for exercises of physics, we take up an eigenvalue problem for the Schrödinger equation with the Woods-Saxon potential, which is very popular in the text books 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 function 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 the Newton s method without recourse to 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 the 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, converges 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 the section 6 is devoted to numerical calculations and results. The 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

8 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 the 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 = 0.67fm, r 0 = 1.27fm, 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 = 940Mev and hc = 197.3Mev 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 can 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 on 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 to 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 to 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 a power series of δx 0. f (x 0 + δx 0 ) = f (x 0 ) + δx 0 f (x 0 ) +. (9) Keeping terms up to the first order of 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

9 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 the S-state with the Woods-Saxon potential In this section we explain briefly the Ishidzu s theory along the lines of his paper [5]. Defining 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 function [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 function [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, now with the help of the relation between the hypergeometric functions [1]: F(s, t; u z) Γ(u)Γ(t s) = Γ(t)Γ(u s) ( z) s F(s, s u + 1; s t + 1 1/z) Γ(u)Γ(s t) + Γ(s)Γ(u t) ( z) t F(t, t u + 1; t s + 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 ). (22) Since d = 0.67fm, R = 7.52fm, and e 1/α =1.34E-5, we keep only the first term in the power series of ( e 1/α ) of the hypergeometric function F. Therfore, the function F in the right-side of equation (22) could be replaced with unity with high precision. Consequently equation (22) becomes If we set 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. θ = arg (23) Γ( 2iκ α) Γ( µ)γ( µ + 1), (24) Volume/Issue/Article Number 3

10 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 2 )π (n = 0, ±1, ±2, ). (25) easily given by argγ(x + iy) = γy tan 1 ( y x ) + m=1[ y m y tan 1 ( x + m )]. (31) 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 (15) and (19), the solutions E n to these transcendental equations (26) are energy eigenvalues. 6 Numerical calculations and the results We consider the Ishidzu s approximate analytical expressions (26) and rewrite them, thanks to the mathematical properties of Gamma function [1], to 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 newly presented form easy to calculate. With the help 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), denoted by argγ(x + iy), is Thanks to this equation, equation (28) is 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 it s summation is carried out from m=1 to m= When α goes to zero in equation (32), equations (27) become 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). Now we can plot arg(e) and h(n) (n = 1, 2, ) against E as in the 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 the 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 the 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 s respectively. And these values can be adopted as the starting values for the Newton s methods. Volume/Issue/Article Number 4

11 [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 the in- h 2 h 2 Table 1: The comparison between the Ishidzu formula and the Runge-Kutta (R-K) method. For the R- ner and let χ(e, c+) denote the wave function at r = c K method we adopt the parameters of the intervals integrated from the outer, and if we write the difference between their Logarithmic derivatives as f (E, c), explained in this section as a = 0.0fm, b = 25.0fm, c = 6.0fm, and N = 500 for convenience. Each of the then it is given by approximate energies is estimated from FIG. 1. The f (E, c) = χ (E, c+) χ(e, c+) χ (E, c ) χ(e, c ). (35) eigenvalues for the 3-dimensional square-well potential are calculated using equation (33) and also calculated Now we can obtain an energy eigenvalue if we from the R-K method. When the R-K method can determine a solution E to f (E, c) = 0, the continuity applied, to avoid the singularity of the square-well of the Logarithmic derivative of the wave func- potential at r = R, the following recipes are taken: tion at r = c. This is an easy task for the Newton s The end point c of the inner interval [0, c] is shifted method. In that case we need to examine the solution from c = R to c = R ; the end point c E to f (E, c) = 0 being free of c. of the outer interval [c, b] is shifted from c = R to c = R Volume/Issue/Article Number 5

12 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 0.67fm (d = r 0.67fm). 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 s using equation (27). 7 Concluding remarks We have demonstrated that the Newton s method is a clearly powerful technique for solving the eigenvalue problems of quantum mechanics and also have endorsed the validity of the Ishizu s analytical solution numerically using the Runge-Kutta method. Although most of books on quantum mechanics are unfamiliar with the 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 the 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 approaches easily. Therefore they may have pedagogical merits for those students and also might be of interest to those instructors who would like to introduce applications of the Newton s method to various fields positively into their courses. Finally, we present the Table 1 to compare the results calculated using the 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 each other. Understandably we use the Newton s method jointly together in order to obtain the convergence value in each individual method. Further more we calculate the energy eigenvalues of the 3-dimensional square-well potential with the same V 0 and R as in (3) for the S-state (l = 0) using the Newton s method. And we include them in the Table 1 for comparison. Besides, by using the 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. 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, [4] A. Erdelyi, W. Magnus, F. oberhettinger, and F. G.Tricomi. Higher Transcendental Functions, volume3, McGraw-Hill Book Co., New York, [5] T. Ishidzu. Analytical solution for s-state with the woods-saxon potentials Prog. Theor. Phys, 40, , Volume/Issue/Article Number 6

13 [6] L. Schiff. Quantum Mechanics 3rd. ed. Mc- GrawHill, New York, Volume/Issue/Article Number 7