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Enumerate the quantum numbers (n, ., and m) for all the
Chapter , Problem 21P(choose chapter or problem)
Enumerate the quantum numbers \((n, \ell, \text { and } m)\) for all the independent states of a hydrogen atom with definite \(E,|\vec{L}|^{2}, \text { and } L_{z}\), up to n = 3. Check that the number of independent states for level n is equal to \(n^2\)
Questions & Answers
QUESTION:
Enumerate the quantum numbers \((n, \ell, \text { and } m)\) for all the independent states of a hydrogen atom with definite \(E,|\vec{L}|^{2}, \text { and } L_{z}\), up to n = 3. Check that the number of independent states for level n is equal to \(n^2\)
ANSWER:Step 1
In each case, we start with the smallest value of n, ℓ, or m possible. Make sure you look over the rules to see how each value was arrived at. ℓ starts at zero and goes to n-1, which is zero since we get 1-1 = 0, when using n = 1. When ℓ = 0, there is only one possible choice for m, which must be zero.
No two electrons can have an identical set of quantum numbers according to the Pauli exclusion principle , so the quantum numbers set limits on the number of electrons which can occupy a given state and therefore give insight into the building up of the periodic table of the elements.
n is principal quantum number
l is orbital quantum number
m is magnetic quantum number