Check the validity of a quantum state, identify atomic orbital designation (1s, 2px, 3dxy…), visualize orbital shape, and understand electron capacity rules. Rigorous quantum mechanical selection rules & Pauli principle.
According to quantum mechanics, every electron in an atom is uniquely described by four quantum numbers: principal (n), azimuthal (l), magnetic (ml), and spin (ms). They emerge from the Schrödinger equation and define the electron's energy, orbital shape, orientation, and intrinsic spin. This calculator rigorously validates any combination based on the Pauli exclusion principle and quantum constraints.
Allowed ranges:
n = 1,2,3,… (energy level)
l = 0,1,2,… , n-1 (s,p,d,f…)
ml = -l, -l+1, …, 0, …, +l
ms = +½ or -½
The azimuthal quantum number l codes the subshell: l=0 → s (sharp), l=1 → p (principal), l=2 → d (diffuse), l=3 → f (fundamental). Each subshell holds up to 2(2l+1) electrons due to spin degeneracy. For a given n, the maximum electrons in the shell is 2n².
| l value | Subshell | ml values | Electron capacity | Orbital shape |
|---|---|---|---|---|
| 0 | s | 0 | 2 | Spherical |
| 1 | p | -1,0,+1 | 6 | Dumbbell |
| 2 | d | -2,-1,0,+1,+2 | 10 | Cloverleaf |
| 3 | f | -3,-2,-1,0,+1,+2,+3 | 14 | Complex |
The Pauli exclusion principle states that no two electrons in an atom can share the same set of four quantum numbers. Our validator checks that n, l, ml, ms satisfy all constraints. In addition, the combination determines the specific atomic orbital (e.g., 2pz corresponds to n=2, l=1, ml=0). The magnetic quantum number distinguishes orbitals within a subshell and influences the angular momentum projection.
Quantum numbers govern selection rules in spectroscopy. For example, NMR exploits the spin quantum number (ms), while atomic emission lines depend on changes in n, l, and ml. Our tool helps students quickly verify allowed transitions and electron configurations before applying Hund’s rules.
The concept originated from Bohr’s model but was refined by Sommerfeld, Pauli, and Schrödinger. The principal quantum number n determines the main energy shell; l accounts for orbital angular momentum; ml defines its orientation; ms was introduced by Uhlenbeck and Goudsmit to explain the Zeeman effect. This calculator implements exactly the rules codified in standard quantum chemistry (Atkins, Griffiths, NIST Atomic Spectroscopy Data).