1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lippmann schwinger Equation, derivation

  1. Aug 26, 2007 #1


    User Avatar
    Science Advisor
    Homework Helper

    [tex] \langle \mathbf{x} \vert \dfrac{1}{E- H_{0} \pm \varepsilon} \vert \mathbf{x'} \rangle = [/tex]

    [tex]\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 [/tex]

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

    Evaluating the parts in the integral:

    [tex]\langle \mathbf{p'} \vert \dfrac{1}{E- H_{0} \pm \varepsilon} \vert \mathbf{p''} \rangle = [/tex]

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

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

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

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

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

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

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

    [tex] \int d^{3}p'' \delta ^{(3)}( \mathbf{p'} - \mathbf{p''}) e^{-i\mathbf{x'}\mathbf{p''}}
    = e^{-i \mathbf{x'} \mathbf{p'}} [/tex] (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.
    Last edited: Aug 26, 2007
  2. jcsd
  3. Aug 26, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    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.
    Last edited: Aug 26, 2007
  4. Aug 26, 2007 #3


    User Avatar
    Science Advisor
    Homework Helper

    Thanx alot, and alos thanx alot for your quick answer!

    But you are telling me that both (1) and (2) is right, how come? This part I do not understand.. =/
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Lippmann schwinger Equation, derivation