Sakurai QM section 5.8

Question

If anyone is familiar with Sakurai's book, specifically section 5.8 on energy shift and decay width, I am stuck and could use some help. I can't see how he derives 5.8.9 (in the revised edition). He starts with (3) (given below), and derives (1) and (2). The equation is as follows:$$(1){\space}{\space}\space{\space}\space{\space}{\dot c_i\over c_i}\approx{{-i\over \hbar}V_{ii}+{\left(-i \over \hbar\right)}^2{{{\left|V_{ii}\right|}^2}\over \eta}+\left({-i\over \hbar}\right) {\sum_{m\not=i}}{{{\left|V_{mi}\right|}^2}\over {(E_i-E_m+i\hbar\eta)}} \over 1-{i \over \hbar}{V_{ii} \over \eta}} $$ $$(2)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\approx{-i\over \hbar}V_{ii}+\left({-i\over \hbar}\right) {\sum_{m\not=i}}{{{\left|V_{mi}\right|}^2}\over {(E_i-E_m+i\hbar\eta)}}$$ where $$(3)\,\,\,\,\,\,\,\,\,c_i(t)=\lim_{\eta\to 0}\left({1-{i\over \hbar\eta}V_{ii}}{e^{{\eta}t}}+{{\left({-i\over \hbar}\right)}^2}{{\left|V_{ii}\right|}^2}{e^{2{\eta}t}\over 2{\eta}^2}+\left({-i\over \hbar}\right){\sum_{m\not=i}{{{{\left|V_{mi}\right|}^2}}\over 2\eta(E_i-E_m+i{\hbar}\eta}}\right)$$First of all, he discards all kinds of terms in the denominator of the first expression, and I can't see how he justifies this, because as ${\eta}\to0$, every term in the expression for $c_i(t)$ blows up (besides the 1 of course). Then he seems to multiply top and bottom by the complex conjugate of the denominator in (1), arriving at (2). He seems to discard a term from the denominator here, which is of second order in V, and says the equation is formally correct up to second order. This doesn't make sense to me! Also, soon after, he has this strange expression with $$\lim_{x\to 0} {1\over(x+iε)}=Pr.(1/x)-iπδ(x)$$ What is this Pr symbol? Any help would be much appreciated!

Answer 1

this does not explain all of it but a bit.

He takes the limit $\eta \to 0$ only after doing this:

$\frac{1}{c_i} \approx \frac{1}{1 - \frac{i}{\hbar} \frac{V_{ii}}{\eta}} \approx 1 +\frac{i}{\hbar} \frac{V_{ii}}{\eta} + ( \frac{i}{\hbar} \frac{V_{ii}}{\eta})^2 +...$ wich you can get by expanding $\frac{1}{1 - \frac{i}{\hbar} \frac{V_{ii}}{\eta}}$ as you can with $ \frac{1}{1-x}=1+x+x^2$

For $\dot c_i$ he takes the derivative and only then takes he the limit $\eta \to 0$ but only in the exponentials (as he says in the book). By which probably a relavie $\eta \to $ small is ment, so it only really kicks in in the exponentials. Then you get :

${-i\over \hbar}V_{ii}+ {\left(-i \over \hbar\right)}^2{{{\left|V_{ii}\right|}^2}\over \eta}+\left({-i\over \hbar}\right){\sum_{m\not=i}}{{{\left|V_{mi}\right|}^2}\over {(E_i-E_m+i\hbar\eta)}}$

After muliplying this by the expression derived for $ { 1 \over c_i}$ , ignoring all terms in V with powers higher then two and cancelling two terms with opposite signs you end up with the expression he gives. So it is as he says accurate up to order two in the potential.