Peskin & Schroeder QFT Born Approximation reference.

[center o bass]

I'm currently teaching myself some QFT trough Peskin and Schroeders Introduction to QFT and I've noticed that in several arguments they rely on appealing to the Born approximation of non-relativistic QM scattering theory. For example on page 121 equation (4.125) they appeal to the scattering amplitude

$$\langle p'|iT|p\rangle = -i \tilde{V}(\vec q) 2\pi \delta(E_\vec{p} - E_\vec{p'}).$$

I thought it might be a good idea to read up on this and I'm trying to find some literature where the scattering amplitude is stated in the form above and explained. I've had no success so far.

Does anyone have any good references?

 

[vanhees71]

This is also known as "Fermi's Golden Rule". You find it in all textbooks on QM where perturbation theory (be it time-independent or time-dependent) is treated. You find the path-integral treatment in the first chapter of my QFT manuscript:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

 

[center o bass]

Is it really Fermi's golden rule?
There the square of $\langle k'|V|k\rangle$ enters and I've never seen it being formulated in terms
of the transition operator T.

 

[center o bass]

Indeed I saw the result in your manuscript formulated in terms of the S matrix. I've never seen Fermi's golden rule stated in that form. How is the S matrix related to T? Do you have any references where the same form is derived in a non integral treatment?

 

[vanhees71]

The scattering operator is related with the transfer operator by
$$\hat{S}=\hat{1}+\mathrm{i} \hat{T}.$$
You find the derivation of the Born series for the S- (or equivalently the T-)matrix in any good quantum mechanics textbook. For scattering theory, I'd recommend

J.J. Sakurai, Modern Quantum Mechanics, Addison Weseley.

 

[center o bass]

I've read most of what seems relevant in J.J Sakurai (Revised Edition) , but I can't seem to make sense of that relation. How does the definition of the T operator

$$V|\Psi^{(+)}\rangle = T |\phi\rangle \ \ \ (7.2.16) \ \ \ page \ 389$$

correspond with the relation you stated

$$S = I + iT?$$

Furthermore by using equation (7.2.2) at page 386 togeather with eqn (7.2.19) at page 389 I get something like

$$\langle \vec k' |T |\vec k\rangle = \tilde{V}(\vec q).$$


J.J Sakurai does not even mention the S-matrix.

 

[andrien]

the non relativistic matrix element Fourier transform will correspond to a potential.the transition matrix contains a four delta function so one term which contains three delta function will give that potential and the left one will give contribution to energy conservation.