[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 ?

----------------------------------------------------------------------------------------------------

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

[Artic_counter]

No one ?

[dag12]

I can help. Give me a short while to respond.

[dag12]

Regarding question 1, the problem is that you should be comparing it relative to the energy itself. That is, you should be calculating ΔE_n/E_n, which, as you will see, goes to zero as n→∞, in correspondence to the classical prediction.

Regarding question 2, I was in the process of typing up a solution, but realized that it would be quicker to refer you to an external source. Reference this document:
http://www.umich.edu/~gevalab/Geva/l...1/Chapter4.pdf
page 6.
Note that the product of sines (as given by you) is only one such separable solution to the 3D particle in a box; a product of complex exponentials could also give you the correct answer.

[Artic_counter]

Thanks a lot! I've been spending a lot of time lately on those two thing!