Quantum Chemistry - Particle in a box

Amblambert · Sep 21, 2019

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!

Handwritten notes on lined paper for a particle-in-a-box problem: sketches of the wavefunctions psi_m(x) and |psi_m(x)|^2 for n=5 between walls at 0 and l, the potential V(x) defined piecewise, and the Hamiltonian and momentum operator equations.

Handwritten working on lined notebook paper: momentum operator applied to sqrt(2/l) sin(n pi x / l), then the integral from 0 to l/4 of the conjugate wavefunction times the wavefunction, with a trig identity sin^2(x) = (1 - cos 2x)/2 noted.

Handwritten derivation on lined paper of the integral of (2/l) times sin squared over the interval 0 to l/4, reduced step by step to a probability expression 1/4 minus sin(m pi/2)/(2 pi m), with a small sine-curve sketch and notes D. What happens when n is large.

$Handwritten working on lined paper for a particle-in-a-box probability calculation, with numbered steps P1, P2, P3 showing fractions with pi and m, boxed results, and a final line listing the options as 1/4 and (pi m plus or minus 2)/(4 pi m).$

jedishrfu · Sep 22, 2019

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 · Sep 23, 2019

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##).

Quantum Chemistry - Particle in a box

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