## PY 502 Homework 4

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## PY 502 Homework 4

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- School:
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- Py 502 - Computatnl Phys

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Homework 4 due Friday October 18 PY 502 Computational Physics Fall 2013 Department of Physics Boston University Instructor Anders Sandvik The deuteron the nucleus of which consists of a neutron and a proton constitutes the simplest composite nuclear system In this assignment you are to solve the radial Schro dinger equation for the bound state of the neutron proton system using a Yukawa potential supplemented by a hard core short range repulsion You will adjust the width and depth parameters of this potential to fit the known binding energy and radius of the deuteron Radial Schro dinger equation for the neutron proton system The wave function for a two particle system with a central potential V r can be written in the form 1 L Lz n x RL n r YL Lz where YL Lz are the spherical harmonics and the radial function satisfies the equation h 2 1 d2 L L 1 h 2 r V r RL n r EL n RL n r 2m r dr2 2mr2 2 where m is the reduced mass m M1 M2 M1 M2 3 Defining the function UL n r rRL n r a simpler radial equation is obtained L L 1 h 2 h 2 d2 V r UL n r EL n UL n r 2m dr2 2mr2 4 5 This is of the same form as the one dimensional Schro dinger equation apart from the fact that there x but here r 0 and the presence of the repulsive centrifugal barrier which effectively contributes to the potential energy For the deuteron L 0 1 and the radial function U r U0 0 r can be written as d2 U r V r 1 dr2 E 6 where we have defined 2m 7 h 2 The deuteron has only one bound state with energy E 2 226 MeV The neutron and proton masses are almost equal Mp 1 6726 10 27 kg and Mp 1 6749 10 27 kg Using these values E 1 Actually due to a small non central nuclear force component L is strictly not a conserved quantum number for the deuteron a small amount of L 2 wave function is mixed with the L 0 state We will here neglect this 1 we get 5 3667 1028 m 2 In order not to have to work with the very small numerical values corresponding to the short inter nuclear distances expressed in meters we change the unit of length from m to fm 1 fm 10 15 m in Eq 6 leading to d2 U r V r 1 U r 0 053667 8 dr2 E Nuclear potential A description of nuclear systems in terms of point particles governed by static central potentials is not completely correct but nevertheless is important as a first approximation Several types of model potentials are used among them the Yukawa potential VY r V0 e r a r a 9 At very short distances the potential should become strongly repulsive which is not a feature of the Yukawa potential A hard core infinite barrier repulsion can be included to accomplish this Then the full potential is V r for r r0 V r VY r V0 e r a for r r0 r a 10 This potential will be used here The three parameters the hard core radius r0 the range a and the depth parameter V0 can be adjusted so that known properties of the deuteron are reproduced Here we shall consider a simplification fixing the hard core radius at r0 0 1 fm 11 The results are in fact not very sensitive to the exact value of r0 To fix the remaining two parameters we will use the binding energy and the radius of the deuteron The radius is defined in terms of the expectation value of its square 1 hr2 i h r2 i 4 12 Here the factor 1 4 comes from the fact that for two particles of equal mass the distance between them correspond to the diameter of a circular orbit not the radius we can here neglect the small mass difference between the neutron and the proton Experimentally one cannot measure the radius directly different radia can be defined depending on what physical scattering process is measured All of the estimates are however close to r 2 fm which we will use here Programming tasks Write a program that solves the radial wave function written in the form 8 with the given value of For a bound state with a potential decaying exponentially to zero at long distances the asymptotic form of the wave function is given by r 13 U r e r 2 The second boundary condition is simple due to the hard core r0 0 14 In this case it is best to start the integration from the outside at some longest distance rmax from the center where rmax is an input value to be read in by the program and you have to figure out by experimentation what a suitable value is at which the wave function is well approximated by the form 13 The integration is done inward and at the last point r0 the second boundary condition 14 should be satisfied Actually provided that rmax is sufficiently large the initial condition at this distance plays a very minor role i e the resulting wave function in the region where it is large depends very little on it Instead of using 13 for the two starting values U rmax and U rmax r it is therefore also fine to choose two arbitrary preferably small 1 values with U rmax U rmax r Normally when solving the Schro dinger equation we are interested in finding the energy eigenvalues Here we are considering the corresponding inverse problem we know the binding energy E 2 226 MeV givenpin the form of the constant and in the ratio with the potential in Eq 8 and the radius r hr2 i 2 fm We want to find the potential that gives rise to a ground state with this energy and no excited bound states For a given value of the Yukawa range parameter a in Eq 10 your program should extract the ratio V0 E for which a bound state is obtained The energy of the bound state is negative relative to the potential at r which here is 0 and hence the ratio V0 E V0 E has to be positive Using values V0 E 0 V 2 V first search for two values between which U r0 changes sign Then use bisection to find the V0 E for which the boundary condition U r0 0 is satisfied Knowing E 2 226 MeV you then have the potential depth parameter V0 that gives the correct binding energy for the range parameter a used 2 You can then calculate the radius using the wave function corresponding to these parameters according to 12 Here you should keep in mind Eqs 1 and 4 and note that the angular part Y of the wave function is normalized and does not enter explicitly in an expectation value of an operator not involving the angles Write …

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