Usuario:Jorgealda/LippmannSchwinger

La ecuación de Lippmann-Schwinger (que recibe su nombre de Bernard Lippmann y Julian Schwinger​) es una ecuación empleada para describir colisiones y dispersión de partículas en mecánica cuántica. Se puede emplear en la dispersión de moléculas, átomos, fotones o cualquier otra partícula, y se utiliza principalmente en física atómica, molecular, óptica, nuclear, y de partículas, pero también en la dispersión de ondas sísimicas en geofísica. Relaciona la onda dispersada con la interacción (potencial de interacción) que causa la dispersión y permite el cálculo de los parámetros experimentales relevantes (amplitud de dispersión y sección eficaz).

La ecuación más fundamental para describir cualquier fenómeno cuántico, incluyendo la dispersión, es la ecuación de Schrödinger. En un problema físico hay que resolver esta ecuación diferencial teniendo en cuenta las condiciones iniciales y/o de contorno específicas del problema. La ecuación de Lippmann-Schwinger es equivalente a la ecuación de Schrödinger más las condiciones de contorno típicas de los problemas de dispersión. Para incluir las condiciones de contorno, la ecuación de Lippmann-Schwinger se debe escribir como una ecuación integral.​ Para los problemas de dispersión, la ecuación de Lippmann-Schwinger suele resultar más conveniente que la ecuación de Schrödinger original.

La forma general de la ecuación de Lippmann-Schwinger (en realidad, hay dos ecuaciones, una para el signo $+\,$ y otra para el signo $-\,$) es:

$|\psi ^{(\pm )}\rangle =|\phi \rangle +{\frac {1}{E-H_{0}\pm i\epsilon }}V|\psi ^{(\pm )}\rangle .\,$

En estas ecuaciones, $|\psi ^{(+)}\rangle \,$ es la función de onda del sistema completo (los dos sistemas que colisionan considerados en conjunto) en un tiempo infinito antes de la interacción, y $|\psi ^{(-)}\rangle \,$, en un tiempo infinitamente después de la interacción ("la función de onda dispersada"). La energía potencial $V\,$ describe la interacción entre los dos sistemas que colisionan. El hamiltoniano $H_{0}\,$ describe la situación en la que los dos sistemas están infinitamente alejados y no interaccionan. Sus autoestados son $|\phi \rangle \,$ y sus autovalores las energias $E\,$. Finalmente, $i\epsilon \,$ es un detalle técnico matemático necesario para el cálculo de las integrales.

Uso

La ecuación de Lippmann-Schwinger es útil en situaciones que involucran la dispersión de dos cuerpos. Para colisiones de tre o más cuerpos no funciona bien debido a limitaciones matemáticas; en su lugar se puede emplear la ecuación de Fadeev.​ Sin embargo, hay métodos aproximados que pueden reducir un problema de muchos cuerpos a un conjunto de problemas de dos cuerpos en varios casos. Por ejemplo, en una colisión entre un electrón y una molécula puede haber decenas o centenas de partículas involucradas, pero el fenómeno se puede reducir a un problema de dos cuerpos si se describe el potencial de todos los constituyentes de la molécula de forma conjunta en un pseudopotencial.​ En estos casos se pueden usar las ecuaciones de Lippmann–Schwinger.

Derivación

Asumiremos que el hamiltoniano se puede escribir como

$H=H_{0}+V$

donde H y H₀ tienen los mismos autovalores y H₀ es un hamiltoniano libre. Por ejemplo, en mecánica cuántica no relativista, H₀ podría ser

$H_{0}={\frac {p^{2}}{2m}}$.

Intuitivamente V es la energía de interacción del sistema. Esta analogía puede ser algo inexacta, ya que las interacciones normalmente cambian la energía E de los estados estacionarios, pero H y H₀ tienen el mismo espectro E_α. Esto significa que, por ejemplo, un estado ligado que sea autoestado del hamiltoniano completo también debe ser autoestado libre, a diferencia de lo que ocurriría al eliminar todas las interacciones, cuando no habría estados ligados. Por lo tanto se puede considerar H₀ como el hamiltoniano libre para estados ligados, con parámetros efectivos que se determinan mediante las interacciones.

Sea un autoestado de H₀:

$H_{0}|\phi \rangle =E|\phi \rangle$.

Si se incluye la interacción $V\,$, hay que resolver

$\left(H_{0}+V\right)|\psi \rangle =E|\psi \rangle$.

Por la continuidad de los autovalores de la energía, hay que imponer $|\psi \rangle \to |\phi \rangle$ cuando $V\to 0$.

Una solución naïve a la ecuación es

$|\psi \rangle =|\phi \rangle +{\frac {1}{E-H_{0}}}V|\psi \rangle$.

donde la notación 1/A denota la inversa of A. Sin embargo, E − H₀ es singular ya que E es un autovalor de H₀.

La singularidad se elimina de dos maneras distintas haciendo el denominador ligeramente complejo:

$|\psi ^{(\pm )}\rangle =|\phi \rangle +{\frac {1}{E-H_{0}\pm i\epsilon }}V|\psi ^{(\pm )}\rangle$.

Insertando un conjunto completo de estados de partícula libre,

$|\psi ^{(\pm )}\rangle =|\phi \rangle +\int d\beta {\frac {|\phi _{\beta }\rangle }{E-E_{\beta }\pm i\epsilon }}\langle \phi _{\beta }|V|\psi ^{(\pm )}\rangle ,\quad H_{0}|\phi _{\beta }\rangle =E_{\beta }|\phi _{\beta }\rangle$,

la ecuación de Schrödinger se convierte en una ecuación integral. Los estados "entrantes" (+) y "salientes" (−) forman también bases, en el pasado y futuro distantes respectivamente, y tienen la apariencia de estados de partículas libres, pero son autoestados del hamiltoniano completo. Añadiendo un índice, la ecuación resultasta

$|\psi _{\alpha }^{(\pm )}\rangle =|\phi \rangle +\int d\beta {\frac {T_{\beta \alpha }|\phi _{\beta }\rangle }{E-E_{\beta }\pm i\epsilon }},\quad T_{\beta \alpha }=\langle \phi _{\beta }|V|\psi _{\alpha }^{(\pm )}\rangle$.

Methods of solution

From the mathematical point of view the Lippmann–Schwinger equation in coordinate representation is the integral equation of Fredholm type. It can be solved by discretization. Since it is equivalent to the differential time-independent Schrödinger equation with appropriate boundary conditions, it can also be solved by numerical methods for differential equations. In the case of the spherically symmetric potential $V$ it is usually solved by partial wave analysis. For high energies and/or weak potential it can also be solved perturbatively by means of Born series. The method convenient also in the case of many-body physics, like in description of atomic, nuclear or molecular collisions is the method of R-matrix of Wigner and Eisenbud. Another class of methods is based on separable expansion of the potential or Green's operator like the method of continued fractions of Horáček and Sasakawa. Very important class of methods is based on variational principles, for example Schwinger variational principle for example the Schwinger-Lanczos method combining the variational principle of Schwinger with Lanczos algorithm.

Interpretation as in and out states

The S-matrix paradigm

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

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

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

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

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

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 $(\phi ,V\psi ^{\pm })$ 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

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

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

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

See also

• Bethe–Salpeter equation

Bibliography

• Joachain, Charles J. (1983). Quantum collision theory. North Holland. ISBN 0-7204-0294-8. 
• Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison Wesley. ISBN 0-201-53929-2. 
• Weinberg, S. (1995). The Quantum Theory of Fields. Cambridge University Press. ISBN 0-521-67053-5.