Lippmann-Schwinger equation
Encyclopedia
In quantum mechanics
the Lippmann–Schwinger equation (named after Bernard A. Lippmann and Julian Schwinger
, Phys. Rev. 79, p. 469, 1950) is of importance to scattering
theory. The equation is
may be written as
where H and H0 have the same eigenvalues and H0 is a free Hamiltonian. For example in nonrelativistic quantum mechanics H0 may be
Intuitively
is the interaction energy of the system. This analogy is somewhat misleading, as interactions generically change the energy levels E of steady states of the system, but H and H0 have identical spectra Eα. This means that, for example, a bound state that is an eigenstate of the interacting Hamiltonian will also be an eigenstate of the free Hamiltonian. This is in contrast with the Hamiltonian obtained by turning off all interactions, in which case there would be no bound states. Thus one may think of H0 as the free Hamiltonian for the boundstates with effective parameters that are determined by the interactions.
Let there be an eigenstate of
:
Now if we add the interaction
into the mix, we need to solve
Because of the continuity of the energy eigenvalues, we wish that
as 
.
A potential solution to this situation is
However
is singular
since
is an eigenvalue of 
.
As is described below, this singularity is eliminated in two distinct ways by making the denominator slightly complex:
, which was pioneered by John Archibald Wheeler
among others, all physical processes are modeled according to the following paradigm.
One begins with a non-interacting multiparticle state in the distant past. Non-interacting does not mean that all of the forces have been turned off, in which case for example proton
s would fall apart, but rather that there exists an interaction-free Hamiltonian
H0 for the bound states which has the same spectrum as the actual Hamiltonian H. This initial state is referred to as the in state. Intuitively, it consists of bound states that are sufficiently well separated that their interactions with each other are ignored.
The idea is that whatever physical process one is trying to study may be modeled as a scattering
process of these well separated bound states. This process is described by the full Hamiltonian H, but once its over all of the new bound states separate again and one finds a new noninteracting state called the out state. The S-matrix is more symmetric under relativity than the Hamiltonian, because it does not require a choice of time slices to define.
This paradigm allows one to calculate the probabilities of all of the processes that we have observed in 70 years of particle collider experiments with remarkable accuracy. But many interesting physical phenomena do not obviously fit into this paradigm. For example, if one wishes to consider the dynamics inside of a neutron star sometimes one wants to know more than what it will finally decay into. In other words, one may be interested in measurements that are not in the asymptotic future. Sometimes an asymptotic past or future is not even available. For example, it is very possible that there is no past before the big bang
.
In the 1960s, the S-matrix paradigm was elevated by many physicists to a fundamental law of nature. In S-matrix theory
, it was stated that any quantity that one could measure should be found in the S-matrix for some process. This idea was inspired by the physical interpretation that S-matrix techniques could give to Feynman diagrams restricted to the mass-shell, and led to the construction of dual resonance model
s. But it was very controversial, because it denied the validity of quantum field theory
based on local fields and Hamiltonians.
of the full Hamiltonian H are the in and out states. The 
are noninteracting states that resemble the in and out states in the infinite past and infinite future.
is an eigenfunction of the Hamiltonian and so at different times only differs by a phase, and so in particular the physical state does not evolve and so it cannot become noninteracting. This problem is easily circumvented by assembling 
and 
into wavepackets with some distribution g(E) of energies E over a characteristic scale 
. The uncertainty principle
now allows the interactions of the asymptotic states to occur over a timescale
and in particular it is no longer inconceivable that the interactions may turn off outside of this interval. The following argument suggests that this is indeed the case.
Plugging the Lippmann–Schwinger equations into the definitions
and
of the wavepackets we see that, at a given time, the difference between the
and 
wavepackets is given by an integral over the energy E.
approach those of 
at time 
and so the corresponding wavepackets are equal at temporal infinity.
In fact, for very positive times t the
factor in a Schrödinger picture
state forces one to close the contour on the lower half-plane. The pole in the V from the Lippmann–Schwinger equation reflects the time-uncertainty of the interaction, while that in the wavepackets weight function reflects the duration of the interaction. Both of these varieties of poles occur at finite imaginary energies and so are suppressed at very large times. The pole in the energy difference in the denominator is on the upper half-plane in the case of
, and so does not lie inside the integral contour and does not contribute to the 
integral. The remainder is equal to the 
wavepacket. Thus, at very late times 
, identifying 
as the asymptotic noninteracting out state.
Similarly one may integrate the wavepacket corresponding to
at very negative times. In this case the contour needs to be closed over the upper half-plane, which therefore misses the energy pole of 
, which is in the lower half-plane. One then finds that the 
and
wavepackets are equal in the asymptotic past, identifying 
as the asymptotic noninteracting in state.
's as asymptotic states is the justification for the 
in the denominator of the Lippmann–Schwinger equations.
of the ath and bth Heisenberg picture
asymptotic states. One may obtain a formula relating the S-matrix to the potential V using the above contour integral strategy, but this time switching the roles of
and 
. As a result, the contour now does pick up the energy pole. This can be related to the 
's if one uses the S-matrix to swap the two 
's. Identifying the coefficients of the 
's on both sides of the equation one finds the desired formula relating S to the potential
In the Born approximation
, corresponding to first order perturbation theory
, one replaces this last
with the corresponding eigenfunction 
of the free Hamiltonian H0, yielding
which expresses the S-matrix entirely in terms of V and free Hamiltonian eigenfunctions.
These formulas may in turn be used to calculate the reaction rate of the process
, which is equal to 
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...
the Lippmann–Schwinger equation (named after Bernard A. Lippmann and Julian Schwinger
Julian Schwinger
Julian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...
, Phys. Rev. 79, p. 469, 1950) is of importance to 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...
theory. The equation is
Derivation
We will assume that the HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
may be written as
where H and H0 have the same eigenvalues and H0 is a free Hamiltonian. For example in nonrelativistic quantum mechanics H0 may be
Intuitively
is the interaction energy of the system. This analogy is somewhat misleading, as interactions generically change the energy levels E of steady states of the system, but H and H0 have identical spectra Eα. This means that, for example, a bound state that is an eigenstate of the interacting Hamiltonian will also be an eigenstate of the free Hamiltonian. This is in contrast with the Hamiltonian obtained by turning off all interactions, in which case there would be no bound states. Thus one may think of H0 as the free Hamiltonian for the boundstates with effective parameters that are determined by the interactions.Let there be an eigenstate of
:Now if we add the interaction
into the mix, we need to solveBecause of the continuity of the energy eigenvalues, we wish that
as .A potential solution to this situation is
However
is singularMathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
since
is an eigenvalue of .As is described below, this singularity is eliminated in two distinct ways by making the denominator slightly complex:
The S-matrix paradigm
In the S-matrix formulation of particle physicsParticle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, which was pioneered by John Archibald Wheeler
John Archibald Wheeler
John Archibald Wheeler was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission...
among others, all physical processes are modeled according to the following paradigm.
One begins with a non-interacting multiparticle state in the distant past. Non-interacting does not mean that all of the forces have been turned off, in which case for example proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s would fall apart, but rather that there exists an interaction-free Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
H0 for the bound states which has the same spectrum as the actual Hamiltonian H. This initial state is referred to as the in state. Intuitively, it consists of bound states that are sufficiently well separated that their interactions with each other are ignored.
The idea is that whatever physical process one is trying to study may be modeled as a 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...
process of these well separated bound states. This process is described by the full Hamiltonian H, but once its over all of the new bound states separate again and one finds a new noninteracting state called the out state. The S-matrix is more symmetric under relativity than the Hamiltonian, because it does not require a choice of time slices to define.
This paradigm allows one to calculate the probabilities of all of the processes that we have observed in 70 years of particle collider experiments with remarkable accuracy. But many interesting physical phenomena do not obviously fit into this paradigm. For example, if one wishes to consider the dynamics inside of a neutron star sometimes one wants to know more than what it will finally decay into. In other words, one may be interested in measurements that are not in the asymptotic future. Sometimes an asymptotic past or future is not even available. For example, it is very possible that there is no past before the big bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
.
In the 1960s, the S-matrix paradigm was elevated by many physicists to a fundamental law of nature. In S-matrix theory
S-matrix theory
S-matrix theory was a proposal for replacing local quantum field theory as the basic principle of elementary particle physics.It avoided the notion of space and time by replacing it with abstract mathematical properties of the S-matrix...
, it was stated that any quantity that one could measure should be found in the S-matrix for some process. This idea was inspired by the physical interpretation that S-matrix techniques could give to Feynman diagrams restricted to the mass-shell, and led to the construction of dual resonance model
Dual resonance model
In theoretical physics, a dual resonance model arose the early investigation of string theory as an S-matrix theory of the strong interaction....
s. But it was very controversial, because it denied the validity of quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
based on local fields and Hamiltonians.
The connection to Lippmann–Schwinger
Intuitively, the slightly deformed eigenfunctions of the full Hamiltonian H are the in and out states. The are noninteracting states that resemble the in and out states in the infinite past and infinite future.Creating wavepackets
This intuitive picture is not quite right, because is an eigenfunction of the Hamiltonian and so at different times only differs by a phase, and so in particular the physical state does not evolve and so it cannot become noninteracting. This problem is easily circumvented by assembling and into wavepackets with some distribution g(E) of energies E over a characteristic scale . The uncertainty principleUncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
now allows the interactions of the asymptotic states to occur over a timescale
and in particular it is no longer inconceivable that the interactions may turn off outside of this interval. The following argument suggests that this is indeed the case.Plugging the Lippmann–Schwinger equations into the definitions
and
of the wavepackets we see that, at a given time, the difference between the
and wavepackets is given by an integral over the energy E.A contour integral
This integral may be evaluated by defining the wave function over the complex E plane and closing the E contour using a semicircle on which the wavefunctions vanish. The integral over the closed contour may then be evaluated, using the Cauchy integral theorem, as a sum of the residues at the various poles. We will now argue that the residues of approach those of at time and so the corresponding wavepackets are equal at temporal infinity.In fact, for very positive times t the
factor in a Schrödinger pictureSchrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...
state forces one to close the contour on the lower half-plane. The pole in the V from the Lippmann–Schwinger equation reflects the time-uncertainty of the interaction, while that in the wavepackets weight function reflects the duration of the interaction. Both of these varieties of poles occur at finite imaginary energies and so are suppressed at very large times. The pole in the energy difference in the denominator is on the upper half-plane in the case of
, and so does not lie inside the integral contour and does not contribute to the integral. The remainder is equal to the wavepacket. Thus, at very late times , identifying as the asymptotic noninteracting out state.Similarly one may integrate the wavepacket corresponding to
at very negative times. In this case the contour needs to be closed over the upper half-plane, which therefore misses the energy pole of , which is in the lower half-plane. One then finds that the and
wavepackets are equal in the asymptotic past, identifying as the asymptotic noninteracting in state.The complex denominator of Lippmann–Schwinger
This identification of the's as asymptotic states is the justification for the in the denominator of the Lippmann–Schwinger equations.A formula for the S-matrix
The S-matrix S is defined to be the inner product
of the ath and bth Heisenberg picture
Heisenberg picture
In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...
asymptotic states. One may obtain a formula relating the S-matrix to the potential V using the above contour integral strategy, but this time switching the roles of
and . As a result, the contour now does pick up the energy pole. This can be related to the 's if one uses the S-matrix to swap the two 's. Identifying the coefficients of the 's on both sides of the equation one finds the desired formula relating S to the potentialIn the Born approximation
Born approximation
In scattering theory and, in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. Born approximation is named after Max Born, winner of the 1954 Nobel Prize for physics.It is...
, corresponding to first order perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
, one replaces this last
with the corresponding eigenfunction of the free Hamiltonian H0, yieldingwhich expresses the S-matrix entirely in terms of V and free Hamiltonian eigenfunctions.
These formulas may in turn be used to calculate the reaction rate of the process
, which is equal to