(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Probability for a momentum value for particle in a box

1. The problem statement, all variables and given/known data

This is the problem 7.37 from Levine's Quantum Chemistry Textbook.

Show that, for a particle in a one-dimensional box of length [tex]l[/tex], the probability of observing a value of [tex]p_{x}[/tex] between [tex]p[/tex] and [tex]p+dp[/tex] is:

[tex]\frac{4|N|^{2}s^{2}}{l(s^{2}-b_{2})}\left(1-(-1)^{n}cos(bl)\right)dp[/tex]

Where [tex]s\equiv n\pi l^{-1}[/tex] and [tex]b\equiv p\hbar^{-1} [/tex]. The constant [tex]N[/tex] is to be chosen so that the integral from minus infinity to infinity of the previous equation is unity.

2. Relevant equations

Since [tex]p[/tex] has a continue range of eigenvalues, the probability of finding a value of [tex]p[/tex] beetween [tex]p[/tex] and [tex]p+dp[/tex] for a system in the state [tex]\Psi[/tex] is

[tex]|\left\langle g_{p}(x)|\Psi (x,t) \right\rangle |^{2}dp [/tex]

where [tex]g_{p}[/tex] are the eigenfunctions of [tex]\hat{p}[/tex].

For a particle in a box:

[tex]\Psi = \left(\frac{2}{l}\right)^{1/2}sin\left(\frac{n\pi x}{l}\right)[/tex]

and the eigenfunctions of the momentum operator are:

[tex]g=Ne^{ipx/\hbar}[/tex]

3. The attempt at a solution

I made the corresponding substitutions to obtain the integral:

[tex]\int ^{0}_{l}Ne^{-ipx/\hbar}\left(\frac{2}{l}\right)^{1/2}sin\left(\frac{n\pi x}{l}\right)dx[/tex]

Which after computing and using some identities gives:

[tex]\int ^{0}_{l}Ne^{-ipx/\hbar}\left(\frac{2}{l}\right)^{1/2}sin\left(\frac{n\pi x}{l}\right)dx = \frac{\sqrt{2}Ns\left[1-(-1)^{n}e^{-ibl}\right] }{\sqrt{l}\left(s^{2}-b^{2}\right)} [/tex]

Which looks a lot like the square root of the expression that I seek, however I'm stuck at this point. Any help will be appreciated, if more informations is needed please tell me.

Thanks in advance

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Probability for a momentum value for particle in a box

**Physics Forums | Science Articles, Homework Help, Discussion**