Solve Schrodinger equation for an eletron in a box. Why discrete?

[tsuwal]

Homework Statement

An eletron is moving along one axis between x=0 and x=L. It's mass is given by m. We want to know the energy and wave function of its possible states given by the quantic number n.

Show that the solution to the above equation is

En=n^2*h^2/(8*m*L^2)

$\Psi(x)=({2/L})^{1/2}*Sin(n*Pi*x/L)$

Homework Equations

The Attempt at a Solution

I see that it is a second order ordinary differencial equation, I should have no problem solving that. However, I can't fully grasp the quantization thing. I mean, I understand why the wavelength is quantized, because of the refletction on its walls, so the wavelength must be 2L/n. But why does the velocity and the frequency must be quanticized? For the energy to be quantized, the velocity must be quanticized and thus, both the wavelength and the frequency must be quanticized. Right? But why is the frequency quantized?

 

[mfb]

Frequency and energy are related via the time-dependent Schrödinger equation: for static solutions, they are proportional to each other.

 

[vela]

I suggest you actually solve the differential equation. The answers to your questions come out in the math.