Computational Physics - Department of Physics

7.8 Schrödinger’s Equation Through Diagonalization 229 whereuk=u(xk),uk± 1 =u(xk±h)andVk=x^2 k, the given potential. Let us see ...

230 7 Eigensystems 7.8.1 Numerical solution of the Schrödinger equation by diagonalization The algorithm for solving Eq. (7.8) m ...

7.8 Schrödinger’s Equation Through Diagonalization 231 include include include"lib.h" using namespacestd; // output file as glob ...

232 7 Eigensystems (int()(constvoid,constvoid*))comp); // send results to ouput file output(r_min , r_max, max_step, d); delete[ ...

7.8 Schrödinger’s Equation Through Diagonalization 233 There are several features to be noted in this program. The main program ...

234 7 Eigensystems Table 7.2Result for numerically calculated eigenvalues of the one-dimensional harmonic oscillator after three ...

7.9 Exercises 235 IF( interpol > 1 )THEN CALLpolint(hh,value(:,l),interpol,0.d0 ,inp,err2) err1=MAX(err1,err2) WRITE(6,'(D12. ...

236 7 Eigensystems − h ̄^2 2 mα^2 d^2 dρ^2 u(ρ)+ k 2 α (^2) ρ (^2) u(ρ) =E u(ρ). We multiply thereafter with 2 mα^2 /h ̄^2 on bo ...

7.9 Exercises 237 and the non-diagonal matrix element ei=−^1 h^2 . In this case the non-diagonal matrix elements are given by a ...

238 7 Eigensystems b) How many pointsnstepdo you need in order to get the lowest three eigenvalues with four leading digits? Rem ...

7.9 Exercises 239 We want to manipulate this equation further to make it as similar to that in (a) as possible. We definekr= 1 / ...

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Part III Differential Equations ...

Here we discuss both ordinary differential equations and partial differential equations, with a focus on finite difference schem ...

Chapter 8 Differential equations If God has made the world a perfect mechanism, he has at least conceded so much to our imperfec ...

244 8 Differential equations So-called delay differential equations that involve functions of one dependent variable, derivativ ...

8.3 Finite difference methods 245 of neutron stars below, we will need to solve two coupled first-order differential equations, ...

246 8 Differential equations and so forth. If the function is rather well-behaved in the domain [a,b], we can use a fixed step s ...

8.3 Finite difference methods 247 ∆(ti,yi(ti)) =f(ti)+ h 2 d f(ti,yi) dt +O(h^3 ). The second derivative can be rewritten as y′′ ...

248 8 Differential equations As this method is numerically stable, it is often used instead of Euler’s method. Another method wh ...