—
(1=yes, 0=no)
—
-1.5111
eV
1.414214
hbar
18
electrons
—
(1=yes, 0=no)
—
-1.5111
eV
1.414214
hbar
18
electrons
The Quantum Number Calculator validates a set of four quantum numbers and computes key properties of the corresponding atomic orbital. In quantum mechanics, the state of an electron in an atom is completely specified by four quantum numbers: the principal quantum number n, the azimuthal (angular momentum) quantum number l, the magnetic quantum number m_l, and the spin quantum number m_s.
The principal quantum number n (n = 1, 2, 3...) determines the electron's energy shell and overall energy. For hydrogen, the energy is E_n = -13.6/n^2 eV. The azimuthal quantum number l (0 to n-1) determines the orbital shape and angular momentum magnitude: L = hbar * sqrt(l*(l+1)). l = 0 gives s orbitals (spherical), l = 1 gives p orbitals (dumbbell), l = 2 gives d orbitals (cloverleaf), l = 3 gives f orbitals.
The magnetic quantum number m_l (ranging from -l to +l, integer values) determines the orbital's orientation in space and the z-component of angular momentum: L_z = m_l * hbar. The spin quantum number m_s (either +1/2 or -1/2) describes the intrinsic spin angular momentum of the electron.
The Pauli exclusion principle states that no two electrons in an atom can have the same set of all four quantum numbers. This fundamental rule determines the electronic structure of all atoms, the periodic table, chemical bonding, and ultimately all of chemistry and materials science. The maximum number of electrons in shell n is 2*n^2 (accounting for all l, m_l, and m_s values).
This calculator validates whether a given set of quantum numbers is physically allowed, names the corresponding orbital, and provides key computed properties.
Validity conditions: n >= 1 (positive integer); 0 <= l <= n-1 (integer); -l <= m_l <= l (integer); m_s = +1/2 or -1/2. Orbital names: l=0 (s), l=1 (p), l=2 (d), l=3 (f), l=4 (g). Hydrogen energy: E_n = -13.6/n^2 eV. Angular momentum magnitude: L = hbar*sqrt(l(l+1)). Max electrons in shell n: 2*n^2.
An is_valid result of 1 means the quantum numbers are allowed; 0 means they violate quantum mechanical rules. The orbital is designated as nl (e.g., 3p for n=3, l=1). The Hydrogen energy is exact; for multi-electron atoms energy also depends on l due to shielding. Max electrons per shell: n=1: 2, n=2: 8, n=3: 18, n=4: 32.
Inputs
Results
The 3p orbital (n=3, l=1) is valid. The hydrogen 3p energy is -1.511 eV. The orbital angular momentum is sqrt(2)*hbar. Shell n=3 holds up to 18 electrons.
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Results
l=2 with n=2 is invalid because l must be less than n. There is no 2d orbital. The maximum l for n=2 is l=1 (giving s and p subshells only).
n (principal, 1,2,3...), l (azimuthal, 0 to n-1), m_l (magnetic, -l to +l), m_s (spin, +1/2 or -1/2). Together they uniquely specify an electron's quantum state in an atom.
The azimuthal quantum number l represents orbital angular momentum of the electron. The constraint l less than n comes from the mathematics of solving the Schrodinger equation in spherical coordinates for the hydrogen atom.
No two electrons in the same atom can have an identical set of all four quantum numbers. This means each orbital (defined by n, l, m_l) can hold at most two electrons, with opposite spins (+1/2 and -1/2).
Shell n holds 2*n^2 electrons: n=1 holds 2, n=2 holds 8, n=3 holds 18, n=4 holds 32. This follows from summing 2*(2l+1) for l = 0 to n-1.
These are orbital shape designations for l = 0, 1, 2, 3 respectively. s orbitals are spherical, p are dumbbell-shaped, d are cloverleaf, f are complex multi-lobed shapes. The letters historically derive from spectroscopic terms: sharp, principal, diffuse, fundamental.
L_z = m_l * hbar. For a p electron (l=1), L_z can be -hbar, 0, or +hbar. This quantization of angular momentum direction is a uniquely quantum mechanical phenomenon with no classical analogue.
Spin determines how electrons fill orbitals (Hund's rule: maximize parallel spins), the magnetic properties of atoms (paramagnetism vs diamagnetism), and covalent bonding (paired electrons with opposite spins form bonds).
The electron has intrinsic angular momentum (spin) of magnitude hbar*sqrt(3)/2 = hbar*sqrt(s(s+1)) with s=1/2. The z-component is m_s*hbar = ±(1/2)*hbar. Spin is a purely quantum mechanical property with no classical mechanical analogue.
The pure Coulomb potential 1/r has a special symmetry that makes all orbitals with the same n degenerate (same energy). In multi-electron atoms, electron-electron repulsion (shielding) breaks this degeneracy, making s orbitals lower energy than p orbitals in the same shell.
l=4 gives g orbitals and l=5 gives h orbitals. These appear in the n=5 and n=6 shells respectively but are not occupied in any known ground-state atom. They would appear in hypothetical atoms with atomic numbers well beyond the current periodic table.
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