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Propagator for free particle: question about Shankar formula

  1. Nov 28, 2015 #1
    I am hung up on what must be a very elementary matter, but I’m unable to see where I’m wrong. I reference R. Shankar's "Principles of Quantum Mechanics". For the free particle with https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png [Broken], the propagator as derived on p. 146 in Eqn. 4.3.13 (or rather in the equation below, which caters to the degenerate case) for a continuous variable is clearly:
    https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png [Broken]
    where https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png [Broken]and https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png [Broken] are the two eigenvectors corresponding to the eigenvalue E. This differs from the form of https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png [Broken] shown in Exercise 5.1.1 on p. 153. Indeed, the change of variable https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png [Broken] gives:
    https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png [Broken]
    By starting with the propagator equation in the eigenvalue variable (E) I end up with the extra [PLAIN]https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png [Broken] [Broken] factor in my integral when I change the variable of integration to p. Since Shankar started with an integral over the p variable, he got the extra factor of https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png [Broken] in Exercise 5.1.1.

    Can anyone help explain why Shankar's Eqn. 5.1.9 is correct without the extra [PLAIN]https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png [Broken] [Broken] factor? Thanks in advance!
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Nov 28, 2015 #2
    Darn, MS Outlook images don't post. It appeared that they would. Too late to delete.
     
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