Can the Lippmann-Schwinger equation be integrated over negative r values?

  • Thread starter Manojg
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In summary, Sakurai converts from Cartesian to spherical coordinates in his book "Modern Quantum Physics, Revised edition" and changes the integration over "q" from "0 to +infinity" to "-infinity to +infinity". This is possible because the integrand is even and it is a formal way of evaluating the integral. It does not have any physical significance and may lead to a contour integration in the complex r plane.
  • #1
Manojg
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Hi,

I have a simple question.
I am looking at Sakurai's "Modern Quantum Physics, Revised edition" on page 382 where he tries to integrate the Lippmann-Schwinger equation. From equation 7.1.15 to 7.1.16, he converted from Cartesian to spherical coordinate system. After integration over [tex]\phi[/tex] and [tex] cos\theta[/tex], he changed the integration over "q" (which is radius in spherical system) from "0 to +infinity" to "-infinity to +infinity".

One can't change radius from -infinity to +infinity in spherical coordinate, right? Then, how did he get that equation?

Thanks.
 
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  • #2
I don't have the book, but sure you can do that if the integrand is even. Doesn't mean there is anything physical about negative r, it's just a formal way of evaluating the integral. Maybe what comes next is a contour integration in the complex r plane?
 

FAQ: Can the Lippmann-Schwinger equation be integrated over negative r values?

What is the Lippmann-Schwinger equation?

The Lippmann-Schwinger equation is a mathematical formula used in quantum mechanics to describe the scattering of a particle off of a potential. It relates the initial and final states of the particle to the potential and the scattering amplitude.

Who developed the Lippmann-Schwinger equation?

The Lippmann-Schwinger equation is named after two physicists, Walter Lippmann and Julius Schwinger, who independently derived the equation in the 1940s.

What is the importance of the Lippmann-Schwinger equation?

The Lippmann-Schwinger equation is a fundamental equation in quantum mechanics and is used to solve many problems in nuclear, atomic, and molecular physics. It allows scientists to calculate the scattering of particles off of a potential and understand the dynamics of these interactions.

How is the Lippmann-Schwinger equation solved?

The Lippmann-Schwinger equation is a non-linear integral equation and can be solved using various methods, such as the Born series expansion or the Green's function method. It requires advanced mathematical techniques and is often solved numerically.

What are the limitations of the Lippmann-Schwinger equation?

The Lippmann-Schwinger equation is limited to non-relativistic systems and does not take into account the effects of spin or the exchange of identical particles. It also assumes a static potential, which may not accurately describe more complex systems. Additionally, it is a single-particle equation and cannot be used to describe interactions between multiple particles.

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