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

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To find the quantum numbers n_(1), n_(2), and n_(3) for the Schrödinger equation in three dimensions, start by determining the energy levels associated with different quantum states. The process involves selecting initial quantum numbers, calculating the corresponding energy values, and then arranging these states in order of energy. For the fourth excited state, one would calculate energies for combinations of quantum numbers, such as E_(112), E_(121), and E_(211), and continue this pattern. The minimum energy state typically occurs at n_(1)=1, n_(2)=1, n_(3)=1, with subsequent states derived from nearby quantum number combinations. This systematic approach allows for the identification of energy levels in both cubic and non-cubic boxes.
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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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