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Effect of a change of coordinates

  1. Aug 10, 2007 #1
    For a wavefunction given by

    [tex]\psi(x,t) = \sum a_n u_n(x) exp(-i E_n T/\hbar) [/tex] how would you show that a change of coordinates x > x + d does not affect the momentum space wavefunction phi(x) by more than a phase change?
    You get phi(x) by Fourier transforming psi.
    So, I do not see why it would affect psi at all because you are moving the origin d to the left but you are integrating over all pace in the Fourier transform.
     
  2. jcsd
  3. Aug 11, 2007 #2

    malawi_glenn

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    You mean we get phi(p) when doing fourier transformation with respect to p. How is E related to p? Have you tried doing the mathematics or are you just trying to solve it by looking at it? =P
     
  4. Aug 11, 2007 #3

    olgranpappy

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    What you are saying exactly is not clear. Do you want to show that for two functions (of 'x') f and g and their Fourier transforms (functions of 'p') F and G, if f and g obey

    [tex]
    g(x)=f(x+a)
    [/tex]

    then F and G obey

    [tex]
    G(p)=e^{ipa}F(p)\;.
    [/tex]

    Is this what you want to show?
     
  5. Aug 11, 2007 #4
    Exactly! That F and G only differ by a phase factor (so their moduli squared are the same).
    [tex]
    \phi(p,t)
    = \int\psi(x+d,t) e^{-i p x/ \hbar}dx
    = \int\psi(u,t) e^{-i p (u - d)/ \hbar}du
    =e^{i p d/ \hbar} \int\psi(u,t) e^{-i p u/ \hbar}du [/tex]

    I think that rephrasing of the question helped me finish it!
     
    Last edited: Aug 11, 2007
  6. Aug 11, 2007 #5

    olgranpappy

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    I'm glad. Good luck w/ the rest of your work.
     
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