Given that the principal quantum number, n, has a value of k in N, is there a closed form equation for calculating the number of atomic orbitals in terms of k?

ghettoking6q

ghettoking6q

Answered question

2022-08-11

Given that the principal quantum number, n, has a value of k N , is there a closed form equation for calculating the number of atomic orbitals in terms of k?
As an example, for n = 2, the number of atomic orbitals is 4. But can this value be calculated for any n = k?

Answer & Explanation

Jaxson White

Jaxson White

Beginner2022-08-12Added 15 answers

This very much depends on the potential. The example you quote is for hydrogen but for the 3d harmonic oscillator there are 1 2 ( n + 1 ) ( n + 2 ) states.
In the nuclear shell model for instance, there is strictly speaking no degeneracy beyond the 2 + 1 states with angular momentum : the energy depends on n and ℓ and states with the same n but different have different energies.
In the case of the infinite spherical well, there is no limit on the possible values of for a given n, and all these E n , states have different energies.

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