Quantum Chemistry
Quantum Numbers
Four numbers that uniquely identify every electron in an atom
Quantum numbers are four values that fully describe an electron's state in an atom. Principal quantum number n (1, 2, 3...) sets size and energy. Angular momentum l (0 to n-1) sets shape (s, p, d, f). Magnetic m_l (-l to +l) sets orientation. Spin m_s (±½) sets electron spin direction. Pauli exclusion: no two electrons share all four. Together, they explain orbital filling, spectroscopy, magnetism. Originated from Bohr-Sommerfeld model + Schrödinger's equation + Stern-Gerlach (spin discovered 1922).
- Principal n1, 2, 3, ... — sets size and main energy
- Angular momentum l0 to n-1 — shape (s=0, p=1, d=2, f=3)
- Magnetic m_l-l, ..., 0, ..., +l — orientation
- Spin m_s+½ or -½ — intrinsic angular momentum
- Pauli exclusionNo two electrons share all four
- Orbitals per shelln² (e.g., n=3 has 9 orbitals)
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Why quantum numbers matter
- Periodic table. Shape reflects orbital filling.
- Bonding. Valence shell electrons determine reactivity.
- Spectroscopy. Transitions between quantized states.
- Magnetism. Spin determines magnetic properties.
- NMR. Nuclear spin used in MRI.
- Quantum computing. Spin states as qubits.
- Atomic physics. Foundation of structure.
Common misconceptions
- Spin is rotation. Intrinsic property; not literal spinning.
- Same n means same energy. True for H; different in multi-electron atoms.
- m_l 0 means stationary orbital. Just orientation; orbitals all equivalent.
- Quantum numbers are arbitrary. Direct from Schrödinger's equation.
- Pauli is for all particles. Only fermions; bosons can share states.
- Quantum numbers describe trajectory. Describe state, not path.
Frequently asked questions
What are the four quantum numbers?
(1) Principal n (1, 2, 3...): shell, size, energy. Larger n = farther from nucleus, higher energy. (2) Angular momentum l (0 to n-1): subshell shape. l=0: s (sphere). l=1: p (dumbbell). l=2: d (cloverleaf). l=3: f (complex). (3) Magnetic m_l (-l to +l): orientation in space. (4) Spin m_s (±½): intrinsic spin direction.
How are they related to orbital energy?
For hydrogen (one electron): only n matters. E ∝ -1/n². For multi-electron atoms: both n and l matter. Lower l means more penetration to nucleus → lower energy at same n. So 4s (n=4, l=0) is lower than 3d (n=3, l=2) — explains why 4s fills first.
What does the spin quantum number represent?
Intrinsic angular momentum of electron. m_s = +½ ("up") or -½ ("down"). Doesn't come from spatial motion — fundamental property like mass and charge. Discovered 1922 (Stern-Gerlach experiment): silver atoms split into two beams in magnetic field. Spin is responsible for paramagnetism, ferromagnetism, NMR, electron paramagnetic resonance.
Why is Pauli exclusion important?
Prevents all electrons from collapsing to the lowest orbital. Forces electrons into higher orbitals → atomic structure. Without Pauli: all atoms would have ground state 1s^Z (e.g., gold = 1s⁷⁹) — collapsed cloud. With Pauli: orbital ladder, periodic table, all of chemistry. Fundamental quantum mechanical principle for fermions.
How many orbitals does each shell have?
n² orbitals per shell. n=1: 1 orbital (1s). n=2: 4 (2s, 2p×3). n=3: 9 (3s, 3p×3, 3d×5). n=4: 16. Each holds 2 electrons → 2n² electrons per shell maximum: 2, 8, 18, 32 (2nd, 3rd, 4th, 5th rows of periodic table).
How do quantum numbers explain the periodic table?
Periods correspond to filling shells. Period 1: 1s. Period 2: 2s, 2p. Period 3: 3s, 3p. Period 4: 4s, 3d, 4p. Etc. s-block (l=0): groups 1-2. p-block (l=1): groups 13-18. d-block (l=2): transition metals. f-block (l=3): lanthanides and actinides. Shape of periodic table reflects orbital structure.
What determines magnetic properties?
Unpaired electrons (each in separate orbital with same spin) make atom paramagnetic — attracted to magnets. All paired (opposite spins): diamagnetic — slightly repelled. Ferromagnetic (Fe, Ni, Co): unpaired d-electrons + structural alignment. Spin quantum numbers underlie all magnetism.