Lippmann schwinger Equation, derivation

[malawi_glenn]

$$\langle \mathbf{x} \vert \dfrac{1}{E- H_{0} \pm \varepsilon} \vert \mathbf{x'} \rangle =$$

$$\int d^{3}p' \int d^{3}p'' \langle \mathbf{x} \vert \mathbf{p'} \rangle \mathbf{p'} \vert \dfrac{1}{E- H_{0} \pm i\varepsilon} \vert \mathbf{p''} \rangle \langle \mathbf{p''} \vert \mathbf{x'} \rangle$$

Operator, acts to the left in this case.
$$H_0 = \dfrac{\mathbf{p}}{2m}$$

Evaluating the parts in the integral:

$$\langle \mathbf{p'} \vert \dfrac{1}{E- H_{0} \pm \varepsilon} \vert \mathbf{p''} \rangle =$$


$$\dfrac{\delta ^{(3)} (\mathbf{p'} - \mathbf{p''} )}{E- \frac{\mathbf{p'}}{2m} \pm i\varepsilon}$$


$$\langle \mathbf{x} \vert \mathbf{p'} \rangle = \dfrac{e^{i\mathbf{x}\mathbf{p'}}}{(2 \pi \hbar)^{3/2}}$$


$$\langle \mathbf{p''} \vert \mathbf{x'} \rangle = \dfrac{e^{-i\mathbf{x'}\mathbf{p''}}}{(2 \pi \hbar)^{3/2}}$$ (1)

Now this last line is wrong (?), it should be:
$$\langle \mathbf{p''} \vert \mathbf{x'} \rangle = \dfrac{e^{-i\mathbf{x'}\mathbf{p'}}}{(2 \pi \hbar)^{3/2}}$$ (2)
According to Sakurai p381, eq (7.1.14)

the integral should become this one when integrating with respect to p''

$$\int d^{3}p' \dfrac{e^{i \mathbf{p'}( \mathbf{x}- \mathbf{x'})}}{E- \frac{ \mathbf{p'}}{2m} \pm i\varepsilon}$$ (3)

If would have continue with my expression for <p''|x'> (1)

$$\int d^{3}p'' \delta ^{(3)}( \mathbf{p'} - \mathbf{p''}) e^{-i\mathbf{x'}\mathbf{p''}} = e^{-i \mathbf{x'} \mathbf{p'}}$$ (4)

Which yields the same result?

Can someone please give some Ideas on this one.

I am unsure if my expression for <p''|x'> is right, and if it is right, if I get the final result (3), and i (4) is right too.

 

[dextercioby]

Nope, the eq (1) is not wrong and yes, you get the right result after an integration wrt p". So equation (3) is the right one and it follows after using (4) , (1), and the matrix element involving delta.

And of course (4) is right as well, since it's a simple integration using the delta functional.

(2) is wrong and (1) is correct. If you've seen (2) in Sakurai, it must be a typo.

 

[malawi_glenn]

Thanx alot, and alos thanks a lot for your quick answer!

But you are telling me that both (1) and (2) is right, how come? This part I do not understand.. =/