Equivalence of Born and eikonal identities

GarethB
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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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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?
 
Thread 'Need help understanding this figure on energy levels'

Thread 'Need help understanding this figure on energy levels'

This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
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Thread 'Understanding how to "tack on" the time wiggle factor'

Thread 'Understanding how to "tack on" the time wiggle factor'

The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
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