Quantum Number Calculator

Quantum numbers are a compact description of an electron state in an atom. The principal quantum number n identifies the shell and is any positive integer. Larger n values generally correspond to orbitals with greater average distance from the nucleus and, in hydrogen-like atoms, less negative energy. The angular momentum quantum number l identifies the subshell and may be any integer from zero through n minus one. Conventional letters map l values of 0, 1, 2, and 3 to s, p, d, and f orbitals. The magnetic quantum number m_l specifies an orbital orientation within a subshell. It takes integer values from negative l through positive l, including zero. That range contains 2l plus 1 possible orbitals. Each spatial orbital can hold two electrons because the spin quantum number m_s has two allowed projections, positive one-half and negative one-half. The Pauli exclusion principle states that no two electrons in one atom can share the same complete set of four quantum numbers. This calculator first checks all four selection rules. For a valid state, it reports the spectroscopic orbital label, the maximum population of the selected shell, and the maximum population of its subshell. A shell has n squared spatial orbitals and can therefore hold 2n squared electrons. A subshell has 2l plus 1 orbitals and can hold twice that number of electrons. These capacities describe the available states; they do not say how many electrons actually occupy an atom without additional electron-configuration information. The displayed energy uses the simple hydrogen-like expression negative 13.6 electron volts divided by n squared. It is exact only for hydrogen in the nonrelativistic approximation and is a useful baseline for one-electron ions when nuclear charge is included separately. In multi-electron atoms, shielding, penetration, electron repulsion, spin-orbit coupling, and other effects split energies that this elementary model treats as equal. Use the result to learn selection rules and shell structure, not as a high-precision prediction for complex atomic spectra.