Computational Physics - M. Jensen Episode 2 Part 5 pdf

13.4. PHYSICS PROJECTS: BOUND STATES IN MOMENTUM SPACE 249 First we need to evaluate the integral over using e.g., gaussian quadrature. This means that we rewrite an integral like where we have fixed lattice points through the corresponding weights and points . The integral in Eq. (13.48) is rewritten as (13.49) We can then rewrite the SE as (13.50) Using the same mesh points for as we did for in the integral evaluation, we get (13.51) with . This is a matrix eigenvalue equation and if we define an matrix to be (13.52) where is the Kronecker delta, and an vector (13.53) we have the eigenvalue problem (13.54) The algorithm for solving the last equation may take the following form Fix the number of mesh points . Use the function in the program library to set up the weights and the points . Before you go on you need to recall that uses the Legendre polynomials to fix the mesh points and weights. This means that the integral is for the interval [-1,1]. Your integral is for the interval [0, ]. You will need to map the weights from to your interval. To do this, call first , with , . It returns the mesh points and 250 CHAPTER 13. EIGENSYSTEMS weights. You then map these points over to the limits in your integral. You can then use the following mapping and is a constant which we discuss below. Construct thereafter the matrix with We are now ready to obtain the eigenvalues. We need first to rewrite the matrix in tri-diagonal form. Do this by calling the library function tred2. This function returns the vector with the diagonal matrix elements of the tri-diagonal matrix while e are the non-diagonal ones. To obtain the eigenvalues we call the function . On return, the array contains the eigenvalues. If is given as the unity matrix on input, it returns the eigenvectors. For a given eigenvalue , the eigenvector is given by the column in , that is z[][k] in C, or z(:,k) in Fortran 90. The problem to solve 1. Before you write the main program for the above algorithm make a dimensional analysis of Eq. (13.48)! You can choose units so that and are in fm . This is the standard unit for the wave vector. Recall then to insert in the appropriate places. For this case you can set the value of . You could also choose units so that the units of and are in MeV. (we have previously used so-called natural units ). You will then need to multiply with MeVfm to obtain the same units in the expression for the potential. Why? Show that must have units MeV . What is the unit of ? If you choose these units you should also multiply the mesh points and the weights with . That means, set the constant . 2. Write your own program so that you can solve the SE in momentum space. 3. Adjust the value of so that you get close to the experimental value of the binding energy of the deuteron, MeV. Which sign should have? 4. Try increasing the number of mesh points in steps of 8, for example 16, 24, etc and see how the energy changes. Your program returns equally many eigenvalues as mesh points . Only the true ground state will be at negative energy. 13.5. PHYSICS PROJECTS: QUANTUM MECHANICAL SCATTERING 251 13.5 Physics projects: Quantum mechanical scattering We are now going to solve the SE for the neutron-proton system in momentum space for positive energies in order to obtain the phase shifts. In the previous physics project on bound states in momentum space, we obtained the SE in momentum space by Eq. (13.48). was the relative momentum between the two particles. A partial wave expansion was used in order to reduce the problem to an integral over the magnitude of momentum only. The subscript referred therefore to a partial wave with a given orbital momentum . To obtain the potential in momentum space we used the Fourier-Bessel transform (Hankel transform) (13.55) where is the spherical Bessel function. We will just study the case , which means that . For scattering states, , the corresponding equation to solve is the so-called Lippman- Schwinger equation. This is an integral equation where we have to deal with the amplitude (reaction matrix) defined through the integral equation (13.56) where the total kinetic energy of the two incoming particles in the center-of-mass system is (13.57) The symbol indicates that Cauchy’s principal-value prescription is used in order to avoid the singularity arising from the zero of the denominator. We will discuss below how to solve this problem. Eq. (13.56) represents then the problem you will have to solve numerically. The matrix relates to the the phase shifts through its diagonal elements as (13.58) The principal value in Eq. (13.56) is rather tricky to evaluate numerically, mainly since com- puters have limited precision. We will here use a subtraction trick often used when dealing with singular integrals in numerical calculations. We introduce first the calculus relation (13.59) It means that the curve has equal and opposite areas on both sides of the singular point . If we break the integral into one over positive and one over negative , a change of variable allows us to rewrite the last equation as (13.60) 252 CHAPTER 13. EIGENSYSTEMS We can use this to express a principal values integral as (13.61) where the right-hand side is no longer singular at , it is proportional to the derivative , and can be evaluated numerically as any other integral. We can then use the trick in Eq. (13.61) to rewrite Eq. (13.56) as (13.62) Using the mesh points and the weights , we can rewrite Eq. (13.62) as (13.63) This equation contains now the unknowns (with dimension ) and . We can turn Eq. (13.63) into an equation with dimension with a mesh which contains the original mesh points for and the point which corresponds to the energy . Consider the latter as the ’observable’ point. The mesh points become then for and . With these new mesh points we define the matrix (13.64) where is the Kronecker and (13.65) and (13.66) With the matrix we can rewrite Eq. (13.63) as a matrix problem of dimension . All matrices , and have this dimension and we get (13.67) or just (13.68) Since we already have defined and (these are stored as matrices) Eq. (13.68) involves only the unknown . We obtain it by matrix inversion, i.e., (13.69) 13.5. PHYSICS PROJECTS: QUANTUM MECHANICAL SCATTERING 253 Thus, to obtain , we need to set up the matrices and and invert the matrix . To do that one can use the function matinv in the program library. With the inverse , performing a matrix multiplication with results in . With we obtain subsequently the phaseshifts using the relation (13.70) Chapter 14 Differential equations 14.1 Introduction Historically, differential equations have originated in chemistry, physics and engineering. More recently they have also been used widely in medicine, biology etc. In this chapter we restrict the attention to ordinary differential equations. We focus on initial value and boundary value problems and present some of the more commonly used methods for solving such problems numerically. The physical systems which are discussed range from a simple cooling problem to the physics of a neutron star. 14.2 Ordinary differential equations (ODE) In this section we will mainly deal with ordinary differential equations and numerical methods suitable for dealing with them. However, before we proceed, a brief remainder on differential equations may be appropriate. The order of the ODE refers to the order of the derivative on the left-hand side in the equation (14.1) This equation is of first order and is an arbitrary function. A second-order equation goes typically like (14.2) A well-known second-order equation is Newton’s second law (14.3) where is the force constant. ODE depend only on one variable, whereas 255 256 CHAPTER 14. DIFFERENTIAL EQUATIONS partial differential equations like the time-dependent Schrödinger equation (14.4) may depend on several variables. In certain cases, like the above equation, the wave func- tion can be factorized in functions of the separate variables, so that the Schrödinger equa- tion can be rewritten in terms of sets of ordinary differential equations. We distinguish also between linear and non-linear differential equation where e.g., (14.5) is an example of a linear equation, while (14.6) is a non-linear ODE. Another concept which dictates the numerical method chosen for solving an ODE, is that of initial and boundary conditions. To give an example, in our study of neutron stars below, we will need to solve two coupled first-order differential equations, one for the total mass and one for the pressure as functions of and where is the mass-energy density. The initial conditions are dictated by the mass being zero at the center of the star, i.e., when , yielding . The other condition is that the pressure vanishes at the surface of the star. This means that at the point where we have in the solution of the integral equations, we have the total radius of the star and the total mass . These two conditions dictate the solution of the equations. Since the differential equations are solved by stepping the radius from to , so-called one-step methods (see the next section) or Runge-Kutta methods may yield stable solutions. In the solution of the Schrödinger equation for a particle in a potential, we may need to apply boundary conditions as well, such as demanding continuity of the wave function and its derivative. In many cases it is possible to rewrite a second-order differential equation in terms of two first-order differential equations. Consider again the case of Newton’s second law in Eq. (14.3). If we define the position and the velocity as its derivative (14.7) 14.3. FINITE DIFFERENCE METHODS 257 we can rewrite Newton’s second law as two coupled first-order differential equations (14.8) and (14.9) 14.3 Finite difference methods These methods fall under the general class of one-step methods. The algoritm is rather simple. Suppose we have an initial value for the function given by (14.10) We are interested in solving a differential equation in a region in space [a,b]. We define a step by splitting the interval in sub intervals, so that we have (14.11) With this step and the derivative of we can construct the next value of the function at (14.12) and so forth. If the function is rather well-behaved in the domain [a,b], we can use a fixed step size. If not, adaptive steps may be needed. Here we concentrate on fixed-step methods only. Let us try to generalize the above procedure by writing the step in terms of the previous step (14.13) where represents the truncation error. To determine , we Taylor expand our function (14.14) where we will associate the derivatives in the parenthesis with (14.15) We define (14.16) and if we truncate at the first derivative, we have (14.17) 258 CHAPTER 14. DIFFERENTIAL EQUATIONS which when complemented with forms the algorithm for the well-known Euler method. Note that at every step we make an approximation error of the order of , however the total error is the sum over all steps , yielding thus a global error which goes like . To make Euler’s method more precise we can obviously decrease (increase ). However, if we are computing the derivative numerically by e.g., the two-steps formula we can enter into roundoff error problems when we subtract two almost equal numbers . Euler’s method is not recommended for precision calculation, although it is handy to use in order to get a first view on how a solution may look like. As an example, consider Newton’s equation rewritten in Eqs. (14.8) and (14.9). We define an . The first steps in Newton’s equations are then (14.18) and (14.19) The Euler method is asymmetric in time, since it uses information about the derivative at the beginning of the time interval. This means that we evaluate the position at using the velocity at . A simple variation is to determine using the velocity at , that is (in a slightly more generalized form) (14.20) and (14.21) The acceleration is a function of and needs to be evaluated as well. This is the Euler-Cromer method. Let us then include the second derivative in our Taylor expansion. We have then (14.22) The second derivative can be rewritten as (14.23) and we can rewrite Eq. (14.14) as (14.24) [...]... following algorithm for the second-order Runge-Kutta method, RK2 ỉ í (14.40) ẵ ắ with the nal value ỉ ãẵ ắ íã í í à ã ẵ ắà ã ắ ã ầ à (14.41) (14. 42) The difference between the previous one-step methods is that we now need an intermediate step in our evaluation, namely ỉ ỉ ãẵ ắà where we evaluate the derivative This ã ắ 14 .5 ADAPTIVE RUNGE-KUTTA AND MULTISTEP METHODS 26 1 involves more operations, but... l e < < setw ( 1 5 ) < < s e t p r e c i s i o n ( 8 ) < < t ; o f i l e < < setw ( 1 5 ) < < s e t p r e c i s i o n ( 8 ) < < y [ 0 ] ; o f i l e < < setw ( 1 5 ) < < s e t p r e c i s i o n ( 8 ) < < y [ 1 ] ; o f i l e < < setw ( 1 5 ) < < s e t p r e c i s i o n ( 8 ) < < cos ( t ) ; o f i l e < < setw ( 1 5 ) < < s e t p r e c i s i o n ( 8 ) < < 0 5 Ê y [ 0 ] Ê y [ 0 ] + 0 5 Ê y [ 1 ] Ê y [1]... earlier, in certain cases it is possible to rewrite a second-order differential equation as two coupled rst-order differential equations With the position ĩ ỉ and the velocity ỉ ĩ ỉ we can reformulate Newtons equation in the following way à à ĩỉà ỉ (14 .51 ) ắ ẳ ĩỉà ỉà ỉ and ỉà (14. 52 ) We are now going to solve these equations using the Runge-Kutta method to fourth order discussed previously Before... be periodic This means that ĩỉ è ĩỉ (14 .53 ) ắ à ã à with è the period dened as è Observe that è depends only on ắ ẳ ắ ễ à ẹ (14 .54 ) ẹ and not on the amplitude of the solution or the constant In addition to the periodicity test, the total energy has also to be conserved Suppose we choose the initial conditions ĩỉ ẳà ẵ ẹ ỉ ẳà ẳ ẹ ì (14 .55 ) 14.6 PHYSICS EXAMPLES 26 3 ẳ but with a potential energy ẵ ẵ ẳ... 14.3.1 Improvements to Eulers algorithm, higher-order methods The most obvious improvements to Eulers and Euler-Cromers algorithms, avoiding in addition the need for computing a second derivative, is the so-called midpoint method We have then ẵà íềãẵ and yielding ẵà íề ã ắà íềãẵ ẵà íềãẵ ắ ắà ắà íềãẵ ã íề ắà íề ã ề ã ầ ắ à ã ầ ắà (14 .26 ) (14 .27 ) ắ (14 .28 ) ắ ề ã ầ à implying that the local truncation... a global error ẵà ắà íề ã íề ã with second-order accuracy for the position and rst-order accuracy for the velocity However, although these methods yield exact results for constant accelerations, the error increases in general with each time step One method that avoids this is the so-called half-step method Here we dene ắà íềãẵ ắ and ắà íềẵ ắ ã ề ã ầ ắ à (14 .29 ) ẵà íềãẵ ắà ẵà íề ã íềãẵ ắ ã ầ ắ à (14.30)... stable, it is often used instead of Eulers method Another method which one may encounter is the Euler-Richardson method with ắà íềãẵ and ắà íề ã ắ ã ầ ắ à (14. 32) ẵà íềãẵ ắà ẵà íề ã íềãẵ ắ ã ầ ắ à (14.33) ềãẵ 26 0 CHAPTER 14 DIFFERENTIAL EQUATIONS 14.4 More on nite difference methods, Runge-Kutta methods Runge-Kutta (RK) methods are based on Taylor expansion formulae, but yield in general better algorithms...14.3 FINITE DIFFERENCE METHODS 25 9 à à which has a local approximation error ầ and a global error ầ ắ These approximations can be generalized by using the derivative to arbitrary order so that we have í ãẵ í ỉ ỉ ã à í ỉ à ã ỉ í à ã ễẵà ỉ í à ễẵ ễ à ã ầ ễãẵ à (14 . 25 ) These methods, based on higher-order derivatives, are in general not used in numerical computation,... ratio ẹ ẵ ắ ẳ 2 Choose the method you wish to employ in solving the problem In the enclosed program we have chosen the fourth-order Runge-Kutta method Subdivide the time interval ỉ ỉ into a grid with step size ỉ where ặ is the number of mesh points ỉ ặ 3 Calculate now the total energy given by ẳ ẵ ĩỉ ẳàắ ẵ ắ ắ and use this when checking the numerically calculated energy from the Runge-Kutta iterations... intermediate step in the computation of í ãẵ To see this, consider rst the following denitions í ỉ and ỉ íà í ỉà and í ãẵ í ỉ íà ỉ ãẵ ã ỉ (14.34) ỉ (14. 35) ỉ íà ỉ (14.36) To demonstrate the philosophy behind RK methods, let us consider the second-order RK method, RK2 The rst approximation consists in Taylor expanding ỉ í around the center of the integration interval ỉ to ỉ ãẵ , i.e., at ỉ , being the step Using . PHYSICS PROJECTS: QUANTUM MECHANICAL SCATTERING 25 1 13 .5 Physics projects: Quantum mechanical scattering We are now going to solve the SE for the neutron-proton system in momentum space for positive energies in. value problems and present some of the more commonly used methods for solving such problems numerically. The physical systems which are discussed range from a simple cooling problem to the physics of. the problem to an integral over the magnitude of momentum only. The subscript referred therefore to a partial wave with a given orbital momentum . To obtain the potential in momentum space we