Partial-wave analysis, in the context of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, refers to a technique for solving
scattering
Scattering is a term used in physics to describe 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 ...
problems by decomposing each wave into its constituent
angular-momentum components and solving using
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s.
Preliminary scattering theory
The following description follows the canonical way of introducing elementary scattering theory. A steady beam of particles scatters off a spherically symmetric potential
, which is short-ranged, so that for large distances
, the particles behave like free particles. In principle, any particle should be described by a
wave packet
In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diff ...
, but we instead describe the scattering of a
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, th ...
traveling along the ''z'' axis, since wave packets can be expanded in terms of plane waves, and this is mathematically simpler. Because the beam is switched on for times long compared to the time of interaction of the particles with the scattering potential, a steady state is assumed. This means that the stationary Schrödinger equation for the wave function
representing the particle beam should be solved:
:
We make the following
ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
:
:
where
is the incoming plane wave, and
is a scattered part perturbing the original wave function.
It is the asymptotic form of
that is of interest, because observations near the scattering center (e.g. an atomic nucleus) are mostly not feasible, and detection of particles takes place far away from the origin. At large distances, the particles should behave like free particles, and
should therefore be a solution to the free Schrödinger equation. This suggests that it should have a similar form to a plane wave, omitting any physically meaningless parts. We therefore investigate the
plane-wave expansion
In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves:
e^ = \sum_^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\hat \cdot \hat),
where
* is the imaginary unit,
* is a wave vector of length ,
* i ...
:
:
The spherical
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
asymptotically behaves like
:
This corresponds to an outgoing and an incoming spherical wave. For the scattered wave function, only outgoing parts are expected. We therefore expect
at large distances and set the asymptotic form of the scattered wave to
:
where
is the so-called ''scattering amplitude'', which is in this case only dependent on the elevation angle
and the energy.
In conclusion, this gives the following asymptotic expression for the entire wave function:
:
Partial-wave expansion
In case of a spherically symmetric potential
, the scattering wave function may be expanded in
spherical harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
s, which reduce to
Legendre polynomial
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
s because of azimuthal symmetry (no dependence on
):
:
In the standard scattering problem, the incoming beam is assumed to take the form of a plane wave of wave number , which can be decomposed into partial waves using the
plane-wave expansion
In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves:
e^ = \sum_^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\hat \cdot \hat),
where
* is the imaginary unit,
* is a wave vector of length ,
* i ...
in terms of
spherical Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
s and
Legendre polynomial
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
s:
:
Here we have assumed a spherical coordinate system in which the axis is aligned with the beam direction. The radial part of this wave function consists solely of the spherical Bessel function, which can be rewritten as a sum of two
spherical Hankel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
:
:
This has physical significance: asymptotically (i.e. for large ) behaves as and is thus an outgoing wave, whereas asymptotically behaves as and is thus an incoming wave. The incoming wave is unaffected by the scattering, while the outgoing wave is modified by a factor known as the partial-wave
S-matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
element :
:
where is the radial component of the actual wave function. The
scattering phase shift
Scattering is a term used in physics to describe 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 ...
is defined as half of the phase of :
:
If flux is not lost, then , and thus the phase shift is real. This is typically the case, unless the potential has an imaginary absorptive component, which is often used in
phenomenological model
A phenomenological model is a scientific model that describes the empirical relationship of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. In other words, a phenomenological ...
s to simulate loss due to other reaction channels.
Therefore, the full asymptotic wave function is
:
Subtracting yields the asymptotic outgoing wave function:
:
Making use of the asymptotic behavior of the spherical Hankel functions, one obtains
:
Since the
scattering amplitude
In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.differential cross section
In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation (e.g. a particle beam, sound wave, light, or an X-ray) intersects a localized phenomenon (e.g. a particle o ...
is given by
:
This works for any short-ranged interaction. For long-ranged interactions (such as the
Coulomb interaction
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
), the summation over may not converge. The general approach for such problems consist in treating the Coulomb interaction separately from the short-ranged interaction, as the Coulomb problem can be solved exactly in terms of
Coulomb functions
In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confl ...
, which take on the role of the Hankel functions in this problem.
References
*
External links
Partial Wave Analysis for Dummies
Quantum mechanics
Scattering theory
{{quantum-stub