Effect of a change of coordinates

1. Aug 10, 2007

ehrenfest

For a wavefunction given by

$$\psi(x,t) = \sum a_n u_n(x) exp(-i E_n T/\hbar)$$ 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. Aug 11, 2007

malawi_glenn

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

3. Aug 11, 2007

olgranpappy

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

$$g(x)=f(x+a)$$

then F and G obey

$$G(p)=e^{ipa}F(p)\;.$$

Is this what you want to show?

4. Aug 11, 2007

ehrenfest

Exactly! That F and G only differ by a phase factor (so their moduli squared are the same).
$$\phi(p,t) = \int\psi(x+d,t) e^{-i p x/ \hbar}dx = \int\psi(u,t) e^{-i p (u - d)/ \hbar}du =e^{i p d/ \hbar} \int\psi(u,t) e^{-i p u/ \hbar}du$$

I think that rephrasing of the question helped me finish it!

Last edited: Aug 11, 2007
5. Aug 11, 2007