Quantum Mechanical Model & Quantum Numbers From Bohr orbits to quantum orbitals: why the change? Bohr’s model explained hydrogen’s line spectrum by putting electrons in circular orbits with fixed energies. But it failed for multi-electron atoms and could not explain finer details like intensities and splitting patterns. The modern view treats electrons as matter waves. Instead of a sharp path, we talk about a fuzzy ‘cloud’ where the electron is likely to be found. Left: Bohr model shows fixed circular orbits. Right: Quantum model shows diffuse probability clouds (orbitals) — no fixed paths. Electrons orbit the nucleus in fixed circular paths like planets around the Sun. Electrons exist as a probability cloud described by a wave function (orbital). There are no fixed paths in the quantum model. Schrödinger’s wave-mechanical model (1926) Erwin Schrödinger (1926) described electrons as standing matter waves around the nucleus. His equation relates the allowed energies of an electron to a mathematical function called the wave function, denoted by (psi). We do not derive it for NEET; we use its consequences: only certain energies and shapes of electron ‘clouds’ are allowed for each atom. Time-independent Schrödinger equation (qualitative) H is the Hamiltonian operator (total energy). is the wave function; E is the allowed energy for that . Comparing and | | 2 : can be positive/negative or complex; | | 2 is always positive and gives probability density at each point in space. Schematic split-panel diagram on white background. Left panel: smooth sinusoidal-like radial/angular wave function curve labeled Psi (can be +/−). Right panel: corresponding non-negative curve labeled |Psi| 2 (probability density). Include nucleus at center, radial axis, and shaded region showing 90–95% probability boundary. Vector, clean lines, red arrows for key annotations. gpt-image-2 2026-05-26T17:04:15.163Z Meaning of vs | | 2 : - (wave function) is a mathematical object that contains all information about the electron. It can be positive/negative or even complex. - | | 2 (probability density) tells how likely the electron is to be found at a point in space. Higher | | 2 means higher chance. An orbital is the region in space where the probability of finding the electron is high (typically the 90–95% probability region from the full distribution). Probability clouds for s, p, d orbitals: dense areas mean higher | | 2 (higher chance to find the electron). Labels remind the quantum-number links. An orbital is a physical container or hard boundary where the electron always stays. An orbital is not a box. It’s a mathematical probability distribution from . The 90–95% surface is a convention, not a rigid wall. remember Orbital = region with high probability (about 90–95%) of finding the electron. It is defined by the wave function , not a sharp edge. The four quantum numbers: an electron’s address Each electron in an atom is labeled by four quantum numbers: principal ( n ), azimuthal ( l ), magnetic ( m l ), and spin ( m s ). Think of n as the ‘floor’ (shell), l as the ‘room type’ (subshell s/p/d/f), m l as the ‘room orientation’, and m s as the ‘spin direction’ (+1/2 or −1/2). Principal quantum number Shell; size and (for multi‑electron atoms) rough energy level 1, 2, 3, ... Number of orbitals in shell ( n 2 ); max electrons in shell (2n 2) Azimuthal (angular momentum) quantum number Subshell type; shape character (s, p, d, f) 0 to (n−1); l=0(s), 1(p), 2(d), 3(f) Number of orbitals in subshell (2l+1) m l Magnetic quantum number Orientation of orbital within a subshell −l to +l (integer steps) Which of the (2l+1) orbitals is chosen m s Spin quantum number Intrinsic spin of electron +1/2 or −1/2 Spin orientation; max two electrons per orbital with opposite spins Quantum number Four quantum numbers at a glance Symbol Name Physical meaning Allowed values Determines For a given shell n , the allowed subshells have l values from 0 up to (n-1) . Within a subshell l , there are (2l+1) allowed m l values: -l, -(l-1), , 0, , +(l-1), +l . tip Letter mapping for subshells: l = 0, 1, 2, 3 ↔ s, p, d, f. Historically: sharp, principal, diffuse, fundamental. 10 14 l value Subshell letter Number of orbitals (2l+1) Max electrons in subshell 2(2l+1) Subshell labels and capacities Quick counts: orbitals per subshell and per shell Each distinct m l value corresponds to one orbital in that subshell. Sum of all orbitals across subshells in shell n equals n 2 . Two electrons (opposite spins) can occupy each orbital. Multiply orbitals per subshell (2l+1) by 2 electrons per orbital. How many orbitals in 3p? l = 1 ⇒ 2l+1 = 3 orbitals ( m l = −1, 0, +1); max electrons = 6. How many orbitals in n = 3? n 2 = 9 orbitals across 3s, 3p, 3d; max electrons = 2n 2 = 18. How many electrons fit in 4f? l = 3 ⇒ 2(2l+1) = 14 electrons; there are 7 f orbitals. Worked mini‑examples 1s 2s, 2p 3s, 3p, 3d 18 4s, 4p, 4d, 4f 16 32 Shell (n) Subshells present Total orbitals (n 2) Max electrons (2n 2) Shells n = 1–4: subshells, orbitals, and maximum electrons Assigning allowed sets of quantum numbers Rules to build a valid set (n, l, m l, m s) : 1) Choose n = 1, 2, 3, . 2) For that n , pick l from 0 to (n-1) . 3) For that l , pick m l from -l to +l (integers). 4) Pick m s = + 1 2 or - 1 2 . That’s it. If any number lies outside its allowed range, the set is invalid. 1s electron: (n, l, m l, m s) = (1, 0, 0, + 1 2 ) is valid. Also (1, 0, 0, - 1 2 ) is valid (two opposite spins). 2p electron: n=2 l=0,1 . For 2p, l=1 and m l -1,0,+1 . So (2,1,0,+ 1 2 ) is valid. Invalid set: (n, l, m l, m s) = (2, 2, 0, + 1 2 ) is invalid because for n=2 , l cannot be 2 (must be 0 or 1). 3d electron: n=3, l=2, m l -2,-1,0,+1,+2 , m s= 1 2 . Example: (3,2,-1,- 1 2 ) is valid. Examples Do not tie a named real orbital like 3d xy to a single m l value in exams. Real d‑orbitals ( d xy , d xz , d yz , d x 2 - y 2 , d z 2 ) are linear combinations of m l states. Safest NEET answer: for 3d, say n=3, l=2 , and m l can be any of −2, −1, 0, +1, +2; m s= 1 2 unless a specific convention is explicitly asked. neet-alert 2026-05-26T17:04:15.123Z gpt-image-2 Tree diagram across three columns for n=1,2,3. For each n, branch to l values (0 to n−1), then each l branches to m l values (−l...+l), then each m l shows two small nodes for m s = ±1/2. Clean vector style, neutral palette, red arrows for branching, no extra text inside shapes. Quantum-number tree for n = 1–3: show allowed l, then allowed m l , then two m s choices per orbital. How the quantum model connects to technology clinical MRI uses nuclear spin (a quantized property) to produce medical images. Hydrogen nuclei in different tissues respond differently to magnetic fields and radio pulses, letting doctors see soft tissue clearly without surgery. MRI links quantum spin with imaging: a scanner manipulates nuclear spins to build contrast between tissues. Glossary you must own Mathematical function whose square magnitude gives electron probability density. Wave function ( ) Probability density ( | | 2 ) Likelihood per unit volume of finding an electron at a point in space. Region in space with high probability (≈90–95%) of finding an electron; defined by . Orbital Principal quantum number (n) Labels shell; determines size and rough energy of orbital. Labels subshell; relates to shape character (s, p, d, f). Azimuthal quantum number (l) Labels orientation; selects one orbital out of the (2l+1) in a subshell. Magnetic quantum number ( m l ) Spin quantum number ( m s ) Intrinsic electron spin: +1/2 or −1/2. Orbitals with the same energy (e.g., three 2p orbitals in hydrogen). Degenerate orbitals Node Region or surface where =0 and hence | | 2=0 ; electron is not found there. 0–1–2–3 → s–p–d–f. Also remember counts: subshell orbitals = 2l+1; shell orbitals = n 2 ; max e− in shell = 2n 2.