Computational Physics - Department of Physics

182 6 Linear Algebra

DOj=1,n

CALLlu_linear_equation(a,n,indx,y(:,j))

ENDDO

! The original matrix a was destroyed, now we equate it with the inverse y

a=y

DEALLOCATE( y )

END SUBROUTINEmatinv

The Fortran programmatinvreceives as input the same variables as the C++ program and
calls the function for LU decompositionlu_decomposeand the function to solve sets of linear
equationslu_linear_equation. The program listed under programs/chapter4/program1.f9 0
performs the same action as the C++ listed above. In order to compile and link these pro-
grams it is convenient to use a so-calledmakefile. Examples of these are found under the
same catalogue as the above programs.

6.4.4.1 Scattering Equation and Principal Value Integrals viaMatrix Inversion

In quantum mechanics, it is often common to rewrite Schrödinger’s equation in momentum
space, after having made a so-called partial wave expansionof the interaction. We will not go
into the details of these expressions but limit ourselves tostudy the equivalent problem for so-
called scattering states, meaning that the total energy of two particles which collide is larger
than or equal zero. The benefit of rewriting the equation in momentum space, after having
performed a Fourier transformation, is that the coordinatespace equation, being an integro-
differantial equation, is transformed into an integral equation. The latter can be solved by
standard matrix inversion techniques. Furthermore, the results of solving these equation can
be related directly to experimental observables like the scattering phase shifts. The latter tell
us how much the incoming two-particle wave function is modified by a collision. Here we take
a more technical stand and consider the technical aspects ofsolving an integral equation with
a principal value.
For scattering states,E> 0 , 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
R(k,k′)(reaction matrix) defined through the integral equation

Rl(k,k′) =Vl(k,k′)+

2

π

P

∫∞

0

dqq^2 Vl(k,q)

1

E−q^2 /m

Rl(q,k′), (6.23)

where the total kinetic energy of the two incoming particlesin the center-of-mass system is

E=

k^20

m. (6.24)

The symbolPindicates that Cauchy’s principal-value prescription is used in order to avoid
the singularity arising from the zero of the denominator. Wewill discuss below how to solve
this problem. Equation (6.23) represents then the problem you will have to solve numerically.
The interaction between the two particles is given by a partial-wave decomposed version
Vl(k,k′), wherelstands for a quantum number like the orbital momentum. We have assumed
that interaction does not coupled to partial waves with different orbital momenta. The vari-
ableskandk′are the outgoing and incoming relative momenta of the two interacting parti-
cles.
The matrixRl(k,k′)relates to the experimental the phase shiftsδlthrough its diagonal
elements as

Rl(k 0 ,k 0 ) =−

tanδl

mk 0

, (6.25)