- #1
fluidistic
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Homework Statement
Hi guys, I've an integral that poped up in QM, it should be different from 0 (else the particle's position is known with 100% certainty while it should not with the wave function I was given). However I get 0 and I don't see where my mistake(s) is/are.
##<(x-a)^2>=\int _{-\infty } ^\infty (x-a)^2N^2 \exp \{ - \frac{(x-a)^2}{2 \sigma ^2} \}dx##.
2. The attempt at a solution
I make the change of variables ##\alpha = (x-a)^2## so that ##d\alpha = 2(x-a)dx##. So as x goes from - infinity to + infinity, alpha goes from + infinity to + infinity and therefore the integral becomes ##\int _{\infty} ^\infty \frac{\sqrt \alpha }{2}N^2 \exp \{ - \frac{\alpha }{2 \sigma ^2} \}d\alpha =0## because of the limits of integration (they are the same).
I've done the exact same change of variables with another integral that yielded 0 this way and it was correct. However here I must not get 0, but I do.
Where did I go wrong?