Quantum number and energy levels

[Amy B]

how does the quantum number n in the wavefunction equation for a particle in a 1D box lead to increasingly well-separated energy levels?
I know that the separation of energy between the levels is given by ΔE = (2n+1)h^2 / 8mL^2 which means that the higher the n, the greater the energy separation, which explains this mathematically. But I just don't understand the concept behind the idea.

 

[strangerep]

The... "concept behind the idea"??

You need to be clearer on what exactly you don't understand. E.g., do you understand how the $\Delta E$ formula was derived from basic QM principles? Do you understand basic QM principles and math? Or is something else unclear?

 

[Simon Bridge]

Is that a question you have been asked?

It is the other way around: the restriction to a potential as in the 1D box potential is what leads to well-separated energy levels... it is convenient to number them, which is where the "quantum number" comes from. Basically the energy level equation $E_n=$... comes first, and we think, "hey that n makes a handy shorthand for referring to energy levels..."