Quantum Mechanical Model of Atom: Quantum Numbers & Orbitals

What is the Quantum Mechanical Model? — Complete Theory

Bohr’s model — brilliant for hydrogen (§ 2.4) — collapsed under two blows from § 2.5: de Broglie showed the electron is a wave, and Heisenberg forbade knowing its exact path. If you cannot know the path, you cannot draw one. In 1926 Erwin Schrödinger wrote the equation that a wave-electron must obey — Hψ = Eψ — and solving it for the hydrogen atom changed the questions chemistry asks. Not “where is the electron?” but “how likely is the electron here?”

The solution ψ (psi) is the wave function — and it has no direct physical meaning by itself. Its square, |ψ|², is the probability density: large where the electron is likely to be found, small where it is not. Plot the region holding ~90–95% of that probability and you have an orbital — a fuzzy three-dimensional cloud, not a racetrack. Where Bohr had one number (n), the quantum atom needs four quantum numbers to pinpoint one electron — three falling out of the Schrödinger equation, one added by experiment:

Read the l-row carefully — it carries the naming system. l = 0, 1, 2, 3 are written s, p, d, f (from sharp, principal, diffuse, fundamental — old spectroscopy jargon), and an orbital is named n + letter: 1s, 2s, 2p, 3d, 4f. Two legality rules follow instantly: l must satisfy l ≤ n − 1 (so 1p, 2d, 3f do not exist), and m must satisfy |m| ≤ l (a p subshell’s orientations run m = −1, 0, +1 — nothing outside).

Roll this up to whole shells: shell n contains n² orbitals (n = 3 → one 3s + three 3p + five 3d = 9) and holds 2n² electrons (18). And every orbital carries internal architecture — nodes, surfaces of zero probability: radial (spherical) nodes = n − l − 1, angular (planar) nodes = l, and their sum is always n − 1. The split shifts between subshells of the same shell; the total never budges — a bookkeeping identity the exam loves to test.

Now the energy question — and here the quantum model draws a line between two universes. Hydrogen (one electron): energy depends only on n. All nine orbitals of n = 3 — 3s, 3p, 3d — are degenerate (equal energy), exactly as in Bohr’s Eₙ = −13.6/n² eV. Multi-electron atoms: energy depends on n + l. Electrons shield each other, so within a shell, lower l sinks lower: 3s < 3p < 3d. This single difference powers the Aufbau principle — fill orbitals in rising n + l order (ties broken by lower n): 1s → 2s → 2p → 3s → 3p → 4s → 3d → …, the sequence behind every electronic configuration you will ever write.

Two filling laws complete the machinery. Pauli’s exclusion principle: no two electrons in an atom share all four quantum numbers — an orbital (fixed n, l, m) can hold at most two electrons, and they must carry opposite spins. Hund’s rule: in degenerate orbitals, electrons first occupy separate orbitals singly with parallel spins before pairing — which is why nitrogen’s 2p³ is three half-filled orbitals, not one full and one empty. (The spin quantum number itself, remember, is the one the Schrödinger equation never produced — it entered through spectroscopy and keeps half-filled and fully-filled subshells extra stable.)

Probability density vs radial probability. ψ² is the probability density — the probability of finding the electron per unit volume at a point. For a 1s electron, ψ² is largest at the nucleus itself. But the chance of finding the electron at a distance r from the nucleus depends on the whole thin spherical shell at that distance, whose volume grows as 4πr²dr. The radial probability is therefore P(r) = 4πr²ψ²dr. For 1s it is zero at the nucleus, rises to a maximum at r = 0.529 Å — Bohr’s first radius, now the most probable distance rather than a fixed orbit — and then falls away. Each radial node appears as a zero in the curve: 2s (one radial node) shows two peaks, 3s (two nodes) shows three. In general the radial probability curve has (n − l) peaks, and the outermost peak is the tallest.

Exceptional configurations: chromium and copper. The Aufbau order predicts Cr (Z = 24) as [Ar] 3d⁴ 4s² and Cu (Z = 29) as [Ar] 3d⁹ 4s². The observed configurations are Cr: [Ar] 3d⁵ 4s¹ and Cu: [Ar] 3d¹⁰ 4s¹ — one 4s electron moves into 3d so that the d subshell becomes exactly half-filled (d⁵) or completely filled (d¹⁰). The 3d and 4s energies are close, so this small shift is repaid by extra stability, which has two sources.

(1) Symmetry. Half-filled and completely filled subshells have a symmetrical distribution of electrons. Electrons in the same subshell shield one another relatively poorly, so they are held more strongly by the nucleus. (2) Exchange energy. Electrons with parallel spins in degenerate orbitals can exchange positions, and each possible exchange releases energy (exchange energy) that stabilises the arrangement. The number of exchanges is greatest when the subshell is half-filled or completely filled: d⁵ with five parallel spins allows 10 exchanges (5 × 4 / 2), against only 6 for d⁴. This is also the reason behind Hund’s rule.

Solved Examples (Step-by-Step)

Numbers → legality → counts. Jo chain yahan chalti hai, wahi validator me live chalti hai.

Key Formulas & Takeaways

Letters: l = 0, 1, 2, 3 → s, p, d, f  ·  Spin: s = ±½ (experimental)  ·  Orbital size: region of ~90–95% probability  ·  Aufbau ties: equal n + l → lower n fills first (3d before 4p)

1. Orbital ≠ orbit — a probability cloud from |ψ|² replaced Bohr’s forbidden exact path; that conceptual swap is the model’s core.
2. Four numbers, one address — n (shell), l (shape, ≤ n−1), m (orientation, |m| ≤ l), s (spin ±½, experimental).
3. Counts are formulas, not memory — n² orbitals and 2n² electrons per shell; 2l+1 orbitals and 2(2l+1) electrons per subshell; nodes n − l − 1 + l = n − 1.
4. Two energy universes — hydrogen degenerate within n; multi-electron ordered by n + l, with Aufbau, Pauli and Hund running the filling.

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