EPTThe energies of the system are given by:
E[n,m]=E*(n^2+m^2) where n,m are integers (quantum numbers in x and y)
then for 0 < n, m < 6 we had to find the maximum degeneracy of any state of the system, which means that there are more than one way for the system to arrange to obtain the same energy. In other words in how many different ways you can pick n and m to come up with the same energy.
for the example with limitation on n and m, i could simply draw a table with values of n and m varying from 1 to 5 and calculating n^2+m^2 and then look what is the maximum number of a value (which would correspond to an energy) repeating in the table. I found that it's 2 and it happens when we interchange n and m.
the interesting bit, which was not part of my homework, is whether this is true for all n and m. So one could rephrase the question:
n^2+m^2=l^2+k^2
where none of the n, m, l or k are equal and all positive. are there any integer solutions to this?
i'm not too good with number theory so i dont have any good ideas how to tackle this. the only thing i can think of now is making a computer program which would draw a "bigger" table and count the equal energies, but this doesnt sound too rigorous or mathematically beautiful. (it would be impossible to disprove anything, because it would have to terminate at some size of the table)
any ideas?
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