Quantum Chemistry - Particle in a box

[Amblambert]

Homework Statement

For the particle in a box, sketch ψn(x) and ψn2(x) for n = 5. Is ψn(x) an eigenfunction of the momentum operator?

If the state of the system is ψn, what is the probability of finding the particle in the
left quarter of the box between 0 and L/4? What happens when n is large?

Relevant Equations

ψn = sqrt(2/l) sin (npix/l)

Here is my attempt at a solution. The thing I am not sure about is the final result of the Shrodinger equation and the n-values that are offered?

Did I make a math mistake?

Thank you so much for reading through this!

 

[jedishrfu]

Ahh, okay. We prefer that you enter everything using latex (see latex link in my signature). While you probably don't use it, its a good skill to develop and it allows us to borrow your equations to correct something.

Anyway, some other poster here may respond to your question. So hang on for an answer.

Here's some references to check out:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html
https://chem.libretexts.org/Bookshe..._Schrodinger_Equation_and_a_Particle_In_a_Box

 

[William Crawford]

Apply l'Hospital's rule to deduce that
\begin{align*}
\lim_{m\rightarrow\infty}\frac{m\pi \pm 2}{4m\pi} &= \frac{1}{4}
\end{align*}
and conclude that the probability for finding the particle in the leftmost quarter of the box is $1/4$ at high energies (i.e. large $m$).