- #1
fluidistic
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Homework Statement
A particle of mass m is inside a 1 dimensional "box" of length L such that it's restricted to move between ##x=-L/2## and ##x=L/2## where the potential vanishes.
1)Determine the eigenvalues ##E_n## and the eigenfunctions ##\psi _n## of the Hamiltonian imposing that the eigenfunctions vanish at the extrema of the box.
2)Graph the first 3 eigenfunctions and analyze their nodes.
The problem continues up to part 5) but I'll post the rest only if I get stuck later.
Homework Equations
Schrödinger's equation.
Normalization of the wave function.
The Attempt at a Solution
1)I solved Schrödinger's equation ##\psi ''+ \varepsilon \psi =0## where ##\varepsilon = \frac{2mE}{\hbar ^2}##.
I've reached that the only possible physical solution is when ##\varepsilon >0## in which case ##\psi (x)=A\cos (\sqrt \varepsilon x )+B \sin (\sqrt \varepsilon x )##.
I applied the boundary conditions to that psi of x, it gave me a system of 2 equations:
[tex]A\cos \left ( \frac{L\sqrt \varepsilon }{2} \right )- B \sin \left ( \frac{L\sqrt \varepsilon }{2} \right )=0[/tex] (*)
[tex]A\cos \left ( \frac{L\sqrt \varepsilon }{2} \right )+ B \sin \left ( \frac{L\sqrt \varepsilon }{2} \right )=0[/tex] (**)
I wrote it under matricial form, I've seen that the matrix of the system is not invertible for some values of epsilon, so I've checked out what condition ##\det M =0## gives me, and it gives me the condition that ##\varepsilon _n=\left ( \frac{n\pi}{L} \right ) ^2##. For such values of epsilon, the matrix is not invertible and those epsilon's describe are directly related to the allowed energies such that the psi_n's are not 0. I should mention that ##n \in \mathbb{N}##, ##n>0##.
This gave me that ##E_n= \frac{\hbar^2 n^2 \pi ^2}{2mL^2}##. So far, so good.
Coming back to (*) (or (**) for that matter) and plugging the values of ##\varepsilon _n## I've found, I've reached that for when n is odd, ##\psi _n(x) =A_n \cos \left ( \frac{n\pi x}{L} \right )## while for when n is even, ##\psi _n (x)=B_n \sin \left ( \frac{n\pi x}{L} \right )##.
So I've answered part 1).
Now for part 2), I must normalize, else I can't graph the first ##3 \psi _n##'s. Normalizing implies finding the constants ##A_n##'s and ##B_n##'s.
I've tried to use ##\int _{-L/2}^{L/2} |\Psi(x)|^2dx=1## where ##\Psi (x)## is an infinite linear combinations of the eigenfunctions, i.e. ##\Psi (x)=\sum _{\text{n odd}} A_n \cos \left ( \frac{n\pi x}{L} \right ) - B_{n+1} \sin \left ( \frac{n\pi x}{L} \right ) ## but this lead me to an integral of an infinite series with 3 terms and I see absolutely no simplications.
I'm stuck at finding those constants, i.e. normalizing the wave function. Am I missing something?
Thanks!