Question about finding quantom numbers N_(n) for Schrodinger Eqn in 3D

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SUMMARY

The discussion centers on determining quantum numbers \( n_1, n_2, n_3 \) for the Schrödinger equation in three dimensions, specifically for cubic and non-cubic boxes. The process involves selecting initial quantum numbers, calculating the corresponding energy levels, and arranging them in order of energy. For example, the first excited states include \( E_{112} \), \( E_{121} \), and \( E_{211} \). The minimum quantum numbers are identified as \( n_1 = n_2 = n_3 = 1 \), and subsequent states are derived by incrementing these values systematically.

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I'm using the Modern Physics by Tipler (6th edition) book.
In sec 7.1 it talks about the first excited state being either E_(112 ) E_(121 ) E_(112).

My question is what is the process of finding the n_(1),n_(2),n_(3) quantum numbers ? How i understand you pick random values and from their find the order of energy levels. Can you give the process if would take of finding let's say the 4th excited state for both a cubic and non cubic box.

Thank you for your time.

PS: I apologize if my format is incorrect this is my fist post. The "_" represent subscript.
 
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Well, if the energy dependence on the quantum numbers were somehow more complicated, you could for example calculate the gradient to find some minimum values, and go from there. In this case, the minimum is trivially at 1,1,1. After you have that sorted, then all you can really do is calculate the energies for nearby quantum numbers. So you'd find E111, E112, ... , E122, E212, and so on. Then you just arrange the states according to their energy.
 

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