Lippmann–Schwinger equation

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The Lippmann–Schwinger equation (named after Bernard A. Lippmann and Julian Schwinger[1]) is one of the most used equations to describe particle collisions - or, more precisely, scattering - in quantum mechanics. It may be used in scattering of molecules, atoms, neutrons, photons or any other particles and is important mainly in atomic, molecular, and optical physics, nuclear physics and particle physics. It relates the scattered wave function with the interaction that produces the scattering (the scattering potential) and therefore allows calculation of the relevant experimental parameters (scattering amplitude and cross sections).

The most fundamental equation to describe any quantum phenomenon, including scattering, is the Schrödinger equation. In physical problems, this differential equation must be solved with the input of an additional set of initial and/or boundary conditions for the specific physical system studied. The Lippmann–Schwinger equation is equivalent to the Schrödinger equation plus the typical boundary conditions for scattering problems. In order to embed the boundary conditions, the Lippmann–Schwinger equation must be written as an integral equation.[2] For scattering problems, the Lippmann–Schwinger equation has been proved to be mathematically and intuitively more convenient than the original Schrödinger equation[citation needed].

The Lippmann–Schwinger equation general shape is (in reality, two equations are shown below, one for the  + \, sign and other for the  - \, sign):

 | \psi^{(\pm)} \rangle = | \phi \rangle + \frac{1}{E - H_0 \pm i \epsilon} V |\psi^{(\pm)} \rangle. \,

In the equations above,  | \psi^{(+)} \rangle \, is the wave function of the whole system (the two colliding systems considered as a whole) at an infinite time before the interaction; and  | \psi^{(-)} \rangle \,, at an infinite time after the interaction (the "scattered wave fuction"). The potential energy  V \, describes the interaction between the two colliding systems. The hamiltonian  H_0 \, describes the situation in which the two systems are infinitely far apart and do not interact. Its eigenfunctions are  | \phi \rangle \, and its eigenvalues are the energies  E \,. Finally,  i \epsilon \, is a mathematical technicality necessary for the calculation of the integrals needed to solve the equation and has no physical meaning.

Usage[edit]

The Lippmann–Schwinger equation is useful in a very large number of situations involving two-body scattering. For three or more colliding bodies it does not work well because of mathematical limitations; Faddeev equations may be used instead.[3] However, there are approximations that can reduce a many-body problem to a set of two-body problems in a variety of cases. For example, in a collision between electrons and molecules, there may be tens or hundreds of particles involved. But the phenomenum may be reduced to a two-body problem by describing all the molecule constituent particle potentials together with a pseudopotential.[4] In these cases, the Lippmann–Schwinger equations may be used. Of course, the main motivations of these approaches are also the possibility of doing the calculations with much lower computational efforts.

Derivation[edit]

We will assume that the Hamiltonian may be written as

H = H_0 + V \,

where H and H0 have the same eigenvalues and H0 is a free Hamiltonian. For example in nonrelativistic quantum mechanics H0 may be

H_0 = \frac{p^2}{2m}. \,

Intuitively V \, 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 H_0 \,:

H_0 | \phi \rangle = E | \phi \rangle. \,

Now if we add the interaction  V \, into the mix, we need to solve

\left( H_0 + V \right) | \psi \rangle = E | \psi \rangle. \,

Because of the continuity of the energy eigenvalues, we wish that  | \psi \rangle \to | \phi \rangle \, as  V \to 0 \,.

A potential solution to this situation is

 |\psi \rangle = | \phi \rangle + \frac{1}{E - H_0} V | \psi \rangle. \,

However (E-H_0) is singular since  E is an eigenvalue of  H_0 .

As is described below, this singularity is eliminated in two distinct ways by making the denominator slightly complex:

 | \psi^{(\pm)} \rangle = | \phi \rangle + \frac{1}{E - H_0 \pm i \epsilon} V |\psi^{(\pm)} \rangle. \,

Interpretation as in and out states[edit]

The S-matrix paradigm[edit]

In the S-matrix formulation of particle physics, 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 protons would fall apart, but rather that there exists an interaction-free Hamiltonian H0, for which the bound states have the same energy level 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 it's 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 models. But it was very controversial, because it denied the validity of quantum field theory based on local fields and Hamiltonians.

The connection to Lippmann–Schwinger[edit]

Intuitively, the slightly deformed eigenfunctions  \psi^{(\pm)} of the full Hamiltonian H are the in and out states. The \phi are noninteracting states that resemble the in and out states in the infinite past and infinite future.

Creating wavepackets[edit]

This intuitive picture is not quite right, because  \psi^{(\pm)} is an eigenfunction of the Hamiltonian and so at different times only differs by a phase. Thus, in particular, the physical state does not evolve and so it cannot become noninteracting. This problem is easily circumvented by assembling  \psi^{(\pm)} and  \phi into wavepackets with some distribution g(E) of energies E over a characteristic scale \Delta E. The uncertainty principle now allows the interactions of the asymptotic states to occur over a timescale \hbar/\Delta E 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

 \psi^{(\pm)}_g(t)=\int dE\, e^{-iEt} g(E)\psi^{(\pm)}

and

 \phi_g(t)=\int dE\, e^{-iEt} g(E)\phi

of the wavepackets we see that, at a given time, the difference between the \psi_g(t) and \phi_g(t) wavepackets is given by an integral over the energy E.

A contour integral[edit]

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  \psi^{(\pm)} approach those of  \phi at time t\rightarrow\mp\infty and so the corresponding wavepackets are equal at temporal infinity.

In fact, for very positive times t the e^{-iEt} 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  \psi^{-}, and so does not lie inside the integral contour and does not contribute to the  \psi^{-} integral. The remainder is equal to the \phi wavepacket. Thus, at very late times  \psi^{-}=\phi, identifying  \psi^{-} as the asymptotic noninteracting out state.

Similarly one may integrate the wavepacket corresponding to  \psi^{+} 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  \psi^{+}, which is in the lower half-plane. One then finds that the  \psi^{+} and  \phi wavepackets are equal in the asymptotic past, identifying  \psi^{+} as the asymptotic noninteracting in state.

The complex denominator of Lippmann–Schwinger[edit]

This identification of the \psi's as asymptotic states is the justification for the \pm\epsilon in the denominator of the Lippmann–Schwinger equations.

A formula for the S-matrix[edit]

The S-matrix S is defined to be the inner product

 S_{ab}=(\psi^-_a,\psi^+_b)

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  \psi^+ and  \psi^-. As a result, the contour now does pick up the energy pole. This can be related to the \phi's if one uses the S-matrix to swap the two \psi's. Identifying the coefficients of the \phi's on both sides of the equation one finds the desired formula relating S to the potential

 S_{ab}=\delta(a-b)-2i\pi\delta(E_a-E_b)(\phi_a,V\psi^+_b).

In the Born approximation, corresponding to first order perturbation theory, one replaces this last  \psi^+ with the corresponding eigenfunction  \phi of the free Hamiltonian H0, yielding

 S_{ab}=\delta(a-b)-2i\pi\delta(E_a-E_b)(\phi_a,V\phi_b)\,

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 b\rightarrow a, which is equal to |S_{ab}-\delta_{ab}|^2.\,

Homogenization[edit]

With the use of Green's function, the Lippmann–Schwinger equation has counterparts in homogenization theory (e.g. mechanics, conductivity, permittivity).

See also[edit]

References[edit]

  1. ^ Lippmann, Bernard A.; Schwinger, Julian (1950). Physical Review Letters 79: 469. Bibcode:1950PhRv...79..469L. doi:10.1103/PhysRev.79.469. 
  2. ^ Joachain, Charles J., 1983 page 112
  3. ^ Joachain, Charles J., 1983 page 517
  4. ^ Joachain, Charles J., 1983 page 576

Bibliography[edit]

  • Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison Wesley. ISBN 0-201-53929-2.