Lippmann-Schwinger Equation

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In summary, the Lippmann-Schwinger Equation is a fundamental equation in quantum mechanics that describes the scattering of particles by a potential. It includes an incident wave function, a scattering potential, and an outgoing wave function, and is derived from the Schrödinger equation using a Green's function. The equation is significant for calculating the scattering properties of particles in various physical systems, but it does have some limitations.
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<bra|ket>
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Hi,
I have a less than acceptable grasp of Complex Analysis. I am confused about the derivation of the Lippmann-Schwinger equation. In the energy basis, a singularity is resolved (denominator: E-K) by adding a small complex term and performing integration over the positive reals. Can someone explain how this accomplishes the task of circumventing the singularity?
Thanks
 
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Roughly:
Look at where the denominator makes the expression undefined. It is no longer on the real axis, so integrating along the real axis is no longer problematic.
 

1. What is the Lippmann-Schwinger Equation?

The Lippmann-Schwinger Equation is a fundamental equation in quantum mechanics used to describe the scattering of particles by a potential. It was developed by Walter Lippmann and Julian Schwinger in the 1940s.

2. What are the key components of the Lippmann-Schwinger Equation?

The Lippmann-Schwinger Equation includes an incident wave function, a scattering potential, and an outgoing wave function. These components are used to calculate the scattering amplitude and cross section of a particle.

3. How is the Lippmann-Schwinger Equation derived?

The Lippmann-Schwinger Equation is derived from the Schrödinger equation and involves the use of a Green's function. The equation describes the scattering process as a series of interactions between the incident and outgoing waves and the potential.

4. What is the significance of the Lippmann-Schwinger Equation?

The Lippmann-Schwinger Equation is important in quantum mechanics because it allows us to calculate the scattering properties of particles in the presence of a potential. It is also used in the study of nuclear reactions, atomic collisions, and scattering of particles in condensed matter.

5. Are there any limitations to the Lippmann-Schwinger Equation?

There are some limitations to the Lippmann-Schwinger Equation, as it assumes a time-independent potential and only considers the first order terms in the scattering process. It also does not take into account the effects of relativity and spin. However, it is still a powerful and widely used equation in many areas of physics.

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