76
2 The Quantum Approach to the Two-Body Problem
where V D is a diagonal matrix and T is a full matrix. It is easy to see that T is a
full matrix since T and V D do not commute TV D = V D T (canonically conjugate
variables) and therefore T cannot be diagonal if V D is diagonal. Then all we need
to do is solve the secular equation to find the appropriate eigenfunctions ψ i (x k ) and
eigenenergies E i .
Hψ = Eψ.
(2.148)
It should be noted that each eigenvector ψ i is just the desired wave function evaluated
at the Gaussian quadrature points instead of the coefficients of the basis functions.
In this procedure, we only need to calculate the potential at the grid points which is
very beneficial especially if the potential depends on other coordinates or is difficult
to calculate. It should be noted that this method is equivalent to a basis set method
and how rapidly it converges depends on the basis function used.
2.4.4 The Approximation to the Potential
If we approximate the potential in each interval with a constant value then the solution
is analytical and of trigonometric type. In the classically allowed regions one uses a
linear combination of sin and cos whereas in the classically forbidden region one uses
a linear combination of sinh and cosh. This allows us, as in the case of approximating
the Laplacian, to develop a propagation method based on these analytic properties.
One can also approximate the potential as a linear function and then the appropriate
solutions within an interval are airy functions Ai(r) and Bi(r).
In fact one can use any approximation to the potential for which there are analytic
solutions.
2.5 Numerical Applications
2.5.1 Systems of Linear Algebraic Equations
The general problem of solving a system of linear algebraic equations of the type
Ax = b
(2.149)
where A is the matrix of coefficients, with x being the vector of unknowns and b
the vector of known terms. The following example uses the method of removal in its
simplest form without any optimization
REPEAT irig1 FOR GOING FROM 1 TO n-1
REPEAT FOR irig2 GOING TO BE A n +1 irig1
2 The Quantum Approach to the Two-Body Problem
where V D is a diagonal matrix and T is a full matrix. It is easy to see that T is a
full matrix since T and V D do not commute TV D = V D T (canonically conjugate
variables) and therefore T cannot be diagonal if V D is diagonal. Then all we need
to do is solve the secular equation to find the appropriate eigenfunctions ψ i (x k ) and
eigenenergies E i .
Hψ = Eψ.
(2.148)
It should be noted that each eigenvector ψ i is just the desired wave function evaluated
at the Gaussian quadrature points instead of the coefficients of the basis functions.
In this procedure, we only need to calculate the potential at the grid points which is
very beneficial especially if the potential depends on other coordinates or is difficult
to calculate. It should be noted that this method is equivalent to a basis set method
and how rapidly it converges depends on the basis function used.
2.4.4 The Approximation to the Potential
If we approximate the potential in each interval with a constant value then the solution
is analytical and of trigonometric type. In the classically allowed regions one uses a
linear combination of sin and cos whereas in the classically forbidden region one uses
a linear combination of sinh and cosh. This allows us, as in the case of approximating
the Laplacian, to develop a propagation method based on these analytic properties.
One can also approximate the potential as a linear function and then the appropriate
solutions within an interval are airy functions Ai(r) and Bi(r).
In fact one can use any approximation to the potential for which there are analytic
solutions.
2.5 Numerical Applications
2.5.1 Systems of Linear Algebraic Equations
The general problem of solving a system of linear algebraic equations of the type
Ax = b
(2.149)
where A is the matrix of coefficients, with x being the vector of unknowns and b
the vector of known terms. The following example uses the method of removal in its
simplest form without any optimization
REPEAT irig1 FOR GOING FROM 1 TO n-1
REPEAT FOR irig2 GOING TO BE A n +1 irig1