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

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SUMMARY

The discussion centers on the integration of the Lippmann-Schwinger equation as presented in Sakurai's "Modern Quantum Physics, Revised edition." The participant questions the validity of changing the integration limits for the radius "q" from "0 to +infinity" to "-infinity to +infinity" in spherical coordinates. It is concluded that while this transformation is mathematically permissible for even integrands, it lacks physical significance for negative radius values. The possibility of contour integration in the complex r plane is suggested as a potential method for resolving the issue.

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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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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?
 

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