- #1
Kostik
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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 , 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
where https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png and https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png 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 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 gives:
https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png
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 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 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 factor? Thanks in advance!
https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png
where https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png and https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png 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 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 gives:
https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png
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 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 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 factor? Thanks in advance!
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