The scattering length in
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
describes low-energy
scattering
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
. For potentials that decay faster than
as
, it is defined as the following low-energy
limit:
:
where
is the scattering length,
is the
wave number, and
is the
phase shift
In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a s ...
of the outgoing spherical wave. The elastic
cross section,
, at low energies is determined solely by the scattering length:
:
General concept
When a slow particle scatters off a short ranged scatterer (e.g. an impurity in a solid or a heavy particle) it cannot resolve the structure of the object since its
de Broglie wavelength is very long. The idea is that then it should not be important what precise
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
one scatters off, but only how the potential looks at long length scales. The formal way to solve this problem is to do a
partial wave expansion (somewhat analogous to the
multipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Multipo ...
in
classical electrodynamics), where one expands in the
angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
components of the outgoing wave. At very low energy the incoming particle does not see any structure, therefore to lowest order one has only a spherical outgoing wave, called the s-wave in analogy with the
atomic orbital
In quantum mechanics, an atomic orbital () is a Function (mathematics), function describing the location and Matter wave, wave-like behavior of an electron in an atom. This function describes an electron's Charge density, charge distribution a ...
at angular momentum quantum number ''l''=0. At higher energies one also needs to consider p and d-wave (''l''=1,2) scattering and so on.
The idea of describing low energy properties in terms of a few parameters and symmetries is very powerful, and is also behind the concept of
renormalization
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
.
The concept of the scattering length can also be extended to potentials that decay slower than
as
. A famous example, relevant for proton-proton scattering, is the Coulomb-modified scattering length.
Example
As an example on how to compute the s-wave (i.e. angular momentum
) scattering length for a given potential we look at the infinitely repulsive spherical
potential well
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy ( kinetic energy in the case of a gravitational potential well) because it is cap ...
of radius
in 3 dimensions. The radial
Schrödinger equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
(
) outside of the well is just the same as for a free particle:
:
where the hard core potential requires that the
wave function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
vanishes at
,
.
The solution is readily found:
:
.
Here
and
is the s-wave
phase shift
In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a s ...
(the phase difference between incoming and outgoing wave), which is fixed by the boundary condition
;
is an arbitrary normalization constant.
One can show that in general
for small
(i.e. low energy scattering). The parameter
of dimension length is defined as the scattering length. For our potential we have therefore
, in other words the scattering length for a hard sphere is just the radius. (Alternatively one could say that an arbitrary potential with s-wave scattering length
has the same low energy scattering properties as a hard sphere of radius
.)
To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the
cross section . In
scattering theory
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
one writes the asymptotic wavefunction as (we assume there is a finite ranged scatterer at the origin and there is an incoming plane wave along the
-axis):
:
where
is the
scattering amplitude. According to the probability interpretation of quantum mechanics the
differential cross section is given by
(the probability per unit time to scatter into the direction
). If we consider only s-wave scattering the differential cross section does not depend on the angle
, and the total
scattering cross section is just
. The s-wave part of the wavefunction
is projected out by using the standard expansion of a plane wave in terms of spherical waves and
Legendre polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
:
:
By matching the
component of
to the s-wave solution
(where we normalize
such that the incoming wave
has a prefactor of unity) one has:
:
This gives:
See also
*
Fermi pseudopotential
*
Neutron scattering length
References
*{{cite book , first=L. D. , last=Landau , first2=E. M. , last2=Lifshitz , year=2003 , title=Quantum Mechanics: Non-relativistic Theory , location=Amsterdam , publisher=Butterworth-Heinemann , isbn=0-7506-3539-8
Quantum mechanics
Scattering theory