Kramers-Heisenberg formula
Encyclopedia
The Kramers-Heisenberg dispersion formula is an expression for the cross section
for scattering
of a photon
by an atom
ic electron
. It was derived before the advent of quantum mechanics
by Hendrik Kramers
and Werner Heisenberg
in 1925, based on the correspondence principle
applied to the classical dispersion formula for light. The quantum mechanical derivation was given by Paul Dirac
in 1927.
The Kramers–Heisenberg formula was an important achievement when it was published, explaining the notion of "negative absorption" (stimulated emission
), the Thomas-Reiche-Kuhn sum rule, and inelastic scattering
- where the energy
of the scattered photon may be larger or smaller than that of the incident photon - thereby anticipating the Raman effect.
It represents the probability of the emission of photons of energy
in the
solid angle
(centred in the 
direction), after the excitation of the system with photons of energy 
. 
are the initial, intermediate
and final states of the system with energy
respectively; the delta
function ensures the energy conservation during the whole process.
is the relevant
transition operator.
is the instrinsic linewidth of the intermediate state.
Cross section (physics)
A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....
for scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
of a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
by an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
ic electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
. It was derived before the advent of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
by Hendrik Kramers
Hendrik Anthony Kramers
Hendrik Anthony "Hans" Kramers was a Dutch physicist.-Background and education:...
and Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
in 1925, based on the correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....
applied to the classical dispersion formula for light. The quantum mechanical derivation was given by Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
in 1927.
The Kramers–Heisenberg formula was an important achievement when it was published, explaining the notion of "negative absorption" (stimulated emission
Stimulated emission
In optics, stimulated emission is the process by which an atomic electron interacting with an electromagnetic wave of a certain frequency may drop to a lower energy level, transferring its energy to that field. A photon created in this manner has the same phase, frequency, polarization, and...
), the Thomas-Reiche-Kuhn sum rule, and inelastic scattering
Inelastic scattering
In particle physics and chemistry, inelastic scattering is a fundamental scattering process in which the kinetic energy of an incident particle is not conserved . In an inelastic scattering process, some of the energy of the incident particle is lost or gained...
- where the energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of the scattered photon may be larger or smaller than that of the incident photon - thereby anticipating the Raman effect.
Equation
The Kramers-Heisenberg (KH) formula for second order processes isIt represents the probability of the emission of photons of energy
in thesolid angle
(centred in the direction), after the excitation of the system with photons of energy . are the initial, intermediateand final states of the system with energy
respectively; the deltafunction ensures the energy conservation during the whole process.
is the relevanttransition operator.
is the instrinsic linewidth of the intermediate state.