[College Physics] - Schrödinger's equation

[Artic_counter]

Hi guys, it's me again! Back with some questions! I'm at beginner level quantum theory btw!

My first question (and the one I would like to be answered the most xD) is concerning The Particle in a Box. In this case it's a one dimensional box.

With Schrödinger's equation, we can determine that the energy of a particle for a particular quantum number "n" is :

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

Therefore

En+1-En=(2n-1)*h^2/(8*m*L^2)

Which means that the difference between two Energy levels will always be discrete. That's in contradiction with the correspondance principle.

That fact is screwing with my head. Is there something I'm not understanding ?

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As for my second question, it's quite the challenge...

Here's how it goes:

In three dimensions, Schrödinger's equation for an electron is :

1.PNG

If the electrons are confined in a cube with borders equals to "L", the stationnary wave, solution of the equation is :

2.PNG

where nx, ny and nz are positive integers.

If we impose the condition of periodicity in L so that :

3.PNG

Demonstrate that:

4.PNG

where

5.PNG