Quantum Number Calculator
Enter the principal quantum number n (1–7) to see all allowed azimuthal (l) and magnetic (ml) quantum numbers, the subshell names (s, p, d, f…), electron capacity per subshell, and the hydrogen energy for that shell.
n = 3 • 2n² = 2 × 3² = 18
- 1
Square the shell number
n² = 3 × 3 = 9n² equals the total number of orbitals in shell n. - 2
Shell electron capacity (2n²)
2 × 9 = 18
How does this calculator work?
For shell n: l runs 0..n−1 (subshells s, p, d, f…); ml runs −l..+l for each l; each orbital holds 2 electrons (ms = ±½). Shell capacity = 2n², orbital count = n². Subshell capacity = 2(2l+1): 2, 6, 10, 14… Hydrogen energy = −13.6/n² eV. Multi-electron atoms differ due to electron repulsion and shielding.
Formula
How this is calculated
In quantum mechanics every electron in an atom is described by four quantum numbers. The principal quantum number n (n = 1, 2, 3, …) labels the electron shell and sets the energy in a hydrogen atom via the Bohr formula E = −13.6/n² eV — the energy is negative because the electron is bound. The azimuthal (angular momentum) quantum number l (l = 0, 1, …, n − 1) specifies the subshell shape: l = 0 is s (spherical), l = 1 is p (two-lobed), l = 2 is d, l = 3 is f, and so on. For each l, the magnetic quantum number ml runs from −l to +l in integer steps (giving 2l + 1 values) and selects the spatial orientation of the orbital. The spin quantum number ms = ±½ is not enumerated here as it applies identically to every orbital.
The total number of orbitals in shell n is n² (the sum of (2l+1) for l = 0..n−1). Since the Pauli exclusion principle allows two electrons (ms = ±½) per orbital, the shell capacity is 2n²: 2 for n=1, 8 for n=2, 18 for n=3, and so on. Each subshell holds 2(2l+1) electrons: 2 (s), 6 (p), 10 (d), 14 (f).
The Bohr energy formula E = −13.6/n² eV is exact only for hydrogen and one-electron ions (He⁺, Li²⁺). In multi-electron atoms, electron–electron repulsion and core shielding shift the energies — subshells of the same n may differ in energy (e.g., 3d is higher than 4s in the ground-state filling order of transition metals).
Frequently asked questions
n (principal — shell size and energy), l (azimuthal — orbital shape), ml (magnetic — orbital orientation), ms (spin — ±½). This calculator covers n, l and ml for a selected shell; ms always takes exactly two values (+½ and −½) for every orbital.
The Bohr formula E = −13.6/n² eV assumes a single electron orbiting a bare nucleus. In multi-electron atoms, repulsion between electrons and shielding of the nuclear charge alter the effective potential, so the formula no longer holds — each element has its own experimental energy levels.
Shell n has n subshells (l = 0 to n−1). Subshell l contains 2l+1 orbitals, and each orbital holds two electrons (ms = ±½). Summing 2(2l+1) from l = 0 to n−1 yields 2n² by the arithmetic series formula.
Also known as
TG we-Calculate Editorial Team. (2026). Quantum Number Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/quantum-number-calculator
TG we-Calculate Editorial Team. "Quantum Number Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/quantum-number-calculator.
TG we-Calculate Editorial Team, "Quantum Number Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/quantum-number-calculator
@misc{wecalculate_quantum_number_calculator, title = {Quantum Number Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/quantum-number-calculator}}, year = {2026}, note = {TG we-Calculate} }
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