2.2 Quantum Elastic Scattering
61
or analytical solutions (such as the Born approximation). This type of approximation
falls within the type of semiclassical (SC) approximations that we also shall discuss
shortly in the followings. For further details see refs. [3, 5, 12].
2.3 Realistic Models for Scattering Systems
2.3.1 Continuum Solutions for Hydrogen-Like Atoms E > 0
As already mentioned, with the Coulomb potential we have faced the problem of
dealing for the first time with the solution for E larger than V (r = ∞) that does not
vanish at large distances but tends to behave as a plane wave. Accordingly, energy
values are not discrete (although they are still eigenstates) and can vary continuously
from zero to infinity. At the same time the variables n and ρ introduced in (2.45) are
imaginary and become
n = −
i
√
2E
= −
i
k
and ρ = 2ikr
(2.80)
where k =
√
2E indicates, as usual, the wave number.
In this case the radial eigenfunctions can be formulated as
R kl =
C k
(2l + 1)!
(2kr)
l e
−ikr F(i/k + l + 1, 2l + 2; 2ikr)
(2.81)
where C K is a normalization factor and F(i/k + l + 1, 2l + 2; 2ikr) is the already
mentioned Hypergeometric function (see [7] for its explicit value).
However, in this case, a situation which is not uncommon in quantum treatments,
having obtained a closed form solution is a bit of a Pyrrhic victory. In fact, the
estimated value of the confluent hypergeometric function for arbitrary values of
the arguments is not easy (and sometimes even impractical) to calculate. For this
reason, even when we can give analytic closed form solutions, we often find it more
convenient to determine their value using numerical techniques.
At this point, for obtaining the amplitude of diffusion and thus of total cross
section it is necessary, as previously mentioned, to analyze the asymptotic behavior
of the wave function (2.81). In the case of the Coulomb interaction (rV (r) = 0 as
r → ∞) in which the solution does not tend to Bessel functions at long range, using
the asymptotic expansions of the confluent hypergeometric function (for the related
rather laborious algebra we refer the reader to refs [5, 7] and exercise 111 in Ref.
[13]) you get
(r)
kr→∞
∼
∞
l=0
(2l + 1)i
l e
iδ l
sin(kr +
1
k
ln 2kr − lπ/2 + φ l )
k
P l (cos θ)
(2.82)
61
or analytical solutions (such as the Born approximation). This type of approximation
falls within the type of semiclassical (SC) approximations that we also shall discuss
shortly in the followings. For further details see refs. [3, 5, 12].
2.3 Realistic Models for Scattering Systems
2.3.1 Continuum Solutions for Hydrogen-Like Atoms E > 0
As already mentioned, with the Coulomb potential we have faced the problem of
dealing for the first time with the solution for E larger than V (r = ∞) that does not
vanish at large distances but tends to behave as a plane wave. Accordingly, energy
values are not discrete (although they are still eigenstates) and can vary continuously
from zero to infinity. At the same time the variables n and ρ introduced in (2.45) are
imaginary and become
n = −
i
√
2E
= −
i
k
and ρ = 2ikr
(2.80)
where k =
√
2E indicates, as usual, the wave number.
In this case the radial eigenfunctions can be formulated as
R kl =
C k
(2l + 1)!
(2kr)
l e
−ikr F(i/k + l + 1, 2l + 2; 2ikr)
(2.81)
where C K is a normalization factor and F(i/k + l + 1, 2l + 2; 2ikr) is the already
mentioned Hypergeometric function (see [7] for its explicit value).
However, in this case, a situation which is not uncommon in quantum treatments,
having obtained a closed form solution is a bit of a Pyrrhic victory. In fact, the
estimated value of the confluent hypergeometric function for arbitrary values of
the arguments is not easy (and sometimes even impractical) to calculate. For this
reason, even when we can give analytic closed form solutions, we often find it more
convenient to determine their value using numerical techniques.
At this point, for obtaining the amplitude of diffusion and thus of total cross
section it is necessary, as previously mentioned, to analyze the asymptotic behavior
of the wave function (2.81). In the case of the Coulomb interaction (rV (r) = 0 as
r → ∞) in which the solution does not tend to Bessel functions at long range, using
the asymptotic expansions of the confluent hypergeometric function (for the related
rather laborious algebra we refer the reader to refs [5, 7] and exercise 111 in Ref.
[13]) you get
(r)
kr→∞
∼
∞
l=0
(2l + 1)i
l e
iδ l
sin(kr +
1
k
ln 2kr − lπ/2 + φ l )
k
P l (cos θ)
(2.82)