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## Homework Statement

A particle with mass m is moving on the x-axis and is described by

## \psi_b = \sqrt{b} \cdot e^{-b |x|}##

Find the probability distribution for the particles momentum

## Homework Equations

## \Phi (p)= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \Psi(x,0) \cdot e^{-ipx} dx##

## The Attempt at a Solution

I just inserted ## \Psi(x,0) \ ## and had a go

## = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \sqrt{b} \cdot e^{-b |x|} \cdot e^{-ipx} dx##

move constants out

## = \sqrt{\frac{b}{2 \pi}}\int_{-\infty}^\infty e^{-b |x|} \cdot e^{-ipx} dx##

combine ##e##

## = \sqrt{\frac{b}{2 \pi}}\int_{-\infty}^\infty e^{-b |x| -ipx} dx##

split integral by

## | x| = \begin{cases} \mbox{x,} & \mbox{if } x>0 \\ \mbox{-x,} & \mbox{if } x <0 \end{cases} ##

so we have

## = \sqrt{\frac{b}{2 \pi}} (\int_{-\infty}^0e^{b x -ipx} dx + \int_{0}^\infty e^{-b x -ipx} dx) ##

perform integration

## = \sqrt{\frac{b}{2 \pi}} ([\frac{e^{b x -ipx}}{ip/t - b}]_{-\infty}^0 + [\frac{e^{-b x -ipx}}{ip/t-b}]_0^{\infty}) ##

evaluating these integrals fails i get

## = \sqrt{\frac{b}{2 \pi}} (\frac{1}{ip/t - b} - \frac{1}{ip/t - b}) = 0 ##

Can you spot my mistakes? would love some input.