MATSLISE: Distorted Coulomb problems |
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The Distorted Coulomb potential
Radial Schrödinger equations have the following form:
y'' = ( l*(l+1)/x^2 + V(x) - E)y, x > 0.
The potential is a distorted Coulomb potential, when
V(x) = S(x)/x + R(x),
where S(x) and R(x) are well behaved functions such that
lim S(x) = S0, lim S(x) = Sas,
x->0
x->inf
lim R(x) = R0, lim R(x) = Ras,
x->0
x->inf
where S0, Sas, R0
and Ras
are constants. Specifically,
it is assumed that around the origin S(x)
and R(x)
can be written in polynomial form and that some r_as
does exist
such that
Vas(x) = Sas/x + Ras
is a good approximation of V(x)
for all r > r_as
.
Solving the distorted Coulomb problem with MATSLISE
In order to specify the problem to solve, the user should provide some input:
S(x)
and R(x)
: The potential function (two functions with no singularities)
l
: a real constant. When l
is a positive integer, it is sometimes called
the orbital quantum number.
xmax
:
A positive constant representing the x-value
where the integration interval should be truncated.
When inf
is entered in the xmax
-field,
then the program determines (for every eigenvalue)
where to truncate in order to obtain estimates
for the eigenvalues within the requested accuracy.
(This may take some more time then entering a value 0 < xmax < inf
.)
parameter name(s)
/ parameter value(s)
:
If the user wants to use a parameter in the fields
S(x)
, R(x)
or l
, he must enter
here the name
of the parametervariable and it's value.
These parameter-fields can also be used to
replace rather lengthy subexpressions in the
potential-function by a parameter.
It is possible to define more than one parameter
(separated by a comma). But it is not allowed
to use parameters in the specification of the
values of the parameters.
tol
:
the accuracy
Solving the distorted Coulomb problem means calculating the eigenvalues E
and the associated eigenfunctions y
. This can be done
by clicking the construct-button.
References
[1] L. Gr. Ixaru, H. De Meyer and G. Vanden Berghe, Highly accurate eigenvalues for the distorted Coulomb potential, Phys. Rev. E 61 (2000) 3151-3159.
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