Equivalence of Born and eikonal identities

In summary, the conversation discusses the need to demonstrate the equivalence between the Born and eikonal identities at very high energies, as well as the satisfaction of the optical theorem by the eikonal amplitude. The speaker mentions the use of Euler's theorem and the trigonometric function sinχ=χ in this demonstration. They also express confusion and ask for recommendations on accessible resources for further understanding.
  • #1
GarethB
15
0
I am required to show that
(i)in the upper limit of very high energies, the Born and eikonal identities are identical.
(ii)that the eikonal amplitude satisfies the optical theorem.

Regarding (i) I think it will involve changing from an exponential to a trig(Euler's theorem) but I could be wrong. The textbook says that sinχ=χ (I think that χ is the profile function).

Regarding (ii) I am clueless. I am trying to do postgrad physics after a period of 6 years since undergrad!
Please help!
 
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  • #2
Ok I have just tried what I thought would be right and failed. Can anyone reference me to a text on this stuff that is understandable?
 

1. What is the significance of the Equivalence of Born and eikonal identities?

The Equivalence of Born and eikonal identities is a fundamental concept in quantum mechanics that relates the scattering amplitudes of particles in a field theory to the classical eikonal phase shifts of the particles. This relationship allows for the calculation of scattering amplitudes using classical methods, making it a powerful tool in theoretical physics.

2. How do the Born and eikonal identities differ?

The Born and eikonal identities are two different mathematical expressions that describe the same physical phenomenon. The Born identity relates the scattering amplitudes to the particle's wave function, while the eikonal identity relates the amplitudes to the eikonal phase shift. They are equivalent because they both describe the same physical process, but from different mathematical perspectives.

3. What is the mathematical representation of the Equivalence of Born and eikonal identities?

The Equivalence of Born and eikonal identities is typically represented by the following equation:
S = 1 + i\tau = e^{i\delta}
where S is the scattering amplitude, i is the imaginary unit, tau is the eikonal phase shift, and delta is the phase shift from the Born identity. This equation shows the relationship between the scattering amplitude and the eikonal and Born phase shifts.

4. How does the Equivalence of Born and eikonal identities relate to Feynman diagrams?

The Equivalence of Born and eikonal identities is closely connected to Feynman diagrams, which are a graphical representation of scattering amplitudes in quantum field theory. The eikonal and Born phase shifts can be calculated using Feynman diagrams, and the Equivalence of Born and eikonal identities allows for the translation of these diagrams into classical calculations.

5. Are there any practical applications of the Equivalence of Born and eikonal identities?

Yes, the Equivalence of Born and eikonal identities has many practical applications in theoretical physics. It is used in the study of high energy particle collisions, such as those performed at the Large Hadron Collider, to calculate scattering amplitudes and make predictions about the behavior of particles. It is also used in quantum field theory to simplify calculations and provide a more intuitive understanding of fundamental processes.

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