# Probability distribution momentum for particle

## 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.

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DrClaude
Mentor
$= \sqrt{\frac{b}{2 \pi}} (\int_{-\infty}^0e^{b x -ipx} dx + \int_{0}^\infty e^{-b x -ipx} dx)$
perform integration
$= \sqrt{\frac{b}{2 \pi}} ([\frac{e^{b x -ipx}}{ip/t - b}]_{-\infty}^0 + [\frac{e^{-b x -ipx}}{ip/t-b}]_0^{\infty})$
Check the denominators.

Check the denominators.
Thanks!
I just had a look and i see it should be:
$= \sqrt{\frac{b}{2 \pi}} (\frac{1}{-ip/t + b} - \frac{1}{-ip/t - b})$
Giving me
$= \sqrt{\frac{b}{2 \pi}} (\frac{1}{-ip/t} + \frac{1}{-ip/t } + \frac{1 }{b}-\frac{1}{b})$
$= \sqrt{\frac{b}{2 \pi}} (-2\frac{1}{-ip/t} )$
$= -2 \sqrt{\frac{b}{2 \pi}} \frac{t}{ip}$
Taking the norm squared to get the probability distribution:
$= 4 \frac{b}{2 \pi} \frac{t^2}{p^2}$

Seems legit to me, but i am not sure.

DrClaude
Mentor
$= \sqrt{\frac{b}{2 \pi}} (\frac{1}{-ip/t + b} - \frac{1}{-ip/t - b})$
Giving me
$= \sqrt{\frac{b}{2 \pi}} (\frac{1}{-ip/t} + \frac{1}{-ip/t } + \frac{1 }{b}-\frac{1}{b})$
What is $t$? And $1/(a+b) \neq 1/a + 1/b$.

What is $t$? And $1/(a+b) \neq 1/a + 1/b$.
Oh off course not.. My blunder..
t is time - are you thinking about finding an expression for t and substituting it?

DrClaude
Mentor
t is time - are you thinking about finding an expression for t and substituting it?
No, I just don't understand why are introducing time in the picture. You are Fourier transforming a function from $x$ to $p$, that's it.

No, I just don't understand why are introducing time in the picture. You are Fourier transforming a function from $x$ to $p$, that's it.
Okay this is embarrassing. It was suppose to be a $\hbar$, but when i wrote from my notes to latex i thought it was a $t$

DrClaude
Mentor
Okay this is embarrassing. It was suppose to be a $\hbar$, but when i wrote from my notes to latex i thought it was a $t$
Then it should also appear in the exponential: $e^{-i p x / \hbar}$.

Right, so i have:
$= \sqrt{\frac{b}{2 \pi}} (\frac{1}{-ip/\hbar + b} - \frac{1}{-ip/\hbar - b})$

Guess there's not much to do - you think taking the norm squared here is a reasonable idea?

Thanks for helping me out.

DrClaude
Mentor
Right, so i have:
$= \sqrt{\frac{b}{2 \pi}} (\frac{1}{-ip/\hbar + b} - \frac{1}{-ip/\hbar - b})$

Guess there's not much to do
Find the common denominator and add the two terms together.

Find the common denominator and add the two terms together.
Off course! Thanks for being so patient with me. Now it works out :)