Quantum Mechanics - Question Regarding Eigenenergy

[vladittude0583]

Homework Statement

Consider an electron in a three-dimensional cubic box of side length L_z. The walls of the box are presumed to correspond to infinitely high potentials.

Homework Equations

E_n=((h²)/8ML_z²)(n²) where n corresponds to the principal quantum numbers in their respective axes

The Attempt at a Solution

My only question is if the side lengths of the cubic box are all of length L_z, does that necessarily mean that all of the principal quantum numbers "n" can be written as n_z also or are they all independent of the each other and of the side lengths? Thanks.

 

[diazona]

Each quantum number is independent. The fact that the side lengths are all the same only means that the energy spectrum (the set of allowed energies) is the same in all 3 dimensions. So for example, the fact that the side lengths are the same means that if n_x = n_y, then the same amount of energy is associated with the x dimension as with the y dimension. But there's no reason that the actual values of different quantum numbers need to be the same, even if the sets of possible values are the same.