Quantum Numbers & Subshells — Quick Reference

Subshells and orbital counts

• Subshells: s, p, d, f

• Orbital counts per subshell: s\to 1\,\text{orbital},\quad p\to 3,\quad d\to 5,\quad f\to 7

• Each orbital holds 2 electrons due to spin degeneracy

Quantum numbers: n, l, ml, ms

• Principal quantum number: n=1,2,3,…

• Angular momentum quantum number: l\in{0,1,\dots, n-1}

• Magnetic quantum number: m_l\in{-l,-l+1,\dots, l}

• Spin quantum number: m_s\in{+\tfrac{1}{2}, -\tfrac{1}{2}}

• Mapping: l=0\to s,\ l=1\to p,\ l=2\to d,\ l=3\to f

• For example, if n=3\;\Rightarrow\; l=0,1,2 (s, p, d)

• For each l, possible ml values: if l=2\Rightarrow ml\in{-2,-1,0,1,2}; if l=0\Rightarrow ml=0; if l=3\Rightarrow ml\in{-3,-2,-1,0,1,2,3}

Orbitals per shell and labeling

• The letter denotes l; the number denotes n

• For a given n, the allowed subshells have: l=0,1,\dots, n-1

• In general: m_l values range from -l to +l (step 1)

• Example: for f-subshell, l=3\Rightarrow m_l\in{-3,-2,-1,0,1,2,3}; 4 is not allowed

Electron capacity and counting

• Number of orbitals in shell n: n^2

• Maximum electrons in shell n: 2n^2

• Example: for n=3, orbitals = 9, electrons = 18

Transitions and energy-wavelength relationship

• Longer wavelength corresponds to lower energy transitions: larger \lambda implies smaller \Delta E

• Energy of a photon from a transition: \Delta E = h c / \lambda

• Conversely: \lambda = h c / \Delta E

• Practical note: transitions are between discrete levels (not continuous), with allowed integer changes in quantum numbers