Quantum numbers — n (principal), l (azimuthal), mₗ (magnetic), mₛ (spin)

2.1.5 · Chemistry › Quantum Atomic Structure

Chaar quantum numbers

Figure — Quantum numbers — n (principal), l (azimuthal), mₗ (magnetic), mₛ (spin)

YEH ranges kyun? (First principles se derivation)

Angular part $m_{l}$ ko quantize kaise karta hai: $ϕ$ -part ko satisfy karna hoga $Φ (ϕ) \propto e^{i m_{l} ϕ}$ . Wavefunction single-valued hone ke liye, ek baar ghoomne ( $ϕ \to ϕ + 2 π$ ) ke baad same value return karni chahiye: $e^{i m_{l} (ϕ + 2 π)} = e^{i m_{l} ϕ} ⟹ e^{i 2 π m_{l}} = 1$ Yeh step kyun? $e^{i 2 π m_{l}} = 1$ tabhi true hai jab $m_{l}$ ek integer ho. Yahi integer $m_{l}$ ki origin hai.

$l$ kyun $m_{l}$ ko bound karta hai: Angular momentum ka projection uski magnitude se zyada nahi ho sakta. Kyunki $L_{z} = m_{l} ℏ$ aur $∣ L ∣ = l (l + 1) ℏ$ , hume chahiye $∣ m_{l} ∣ \leq l$ , jo $m_{l} = - l, \dots, + l$ deta hai.

$n$ kyun $l$ ko bound karta hai: Radial equation sirf tabhi well-behaved (finite, normalizable) solutions produce karta hai jab $l \leq n - 1$ . Physically: zyada angular motion "energy budget use kar leta hai," isliye ek given shell $n$ sirf limited orbital shapes host kar sakta hai.

Ek shell mein electrons ginne ka tarika: $states = m_{s} 2 \times \sum_{l = 0}^{n - 1} (2 l + 1) = 2 n^{2}$ Kyun? Sum $\sum_{l = 0}^{n - 1} (2 l + 1) = n^{2}$ (pehle $n$ odd numbers ka sum), spin ke liye 2 se multiply.

Worked examples

Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho electrons ke liye ek badi apartment building hai. n kaun sa floor hai (upar wala floor = wahan pahunchne ke liye zyada energy chahiye). l us floor ke rooms ki shape hai (round rooms, dumbbell rooms, cloverleaf rooms). mₗ woh direction hai jis taraf room point karta hai. mₛ yeh hai ki electron upar ki taraf munh karke sota hai ya neeche ki taraf. Magic rule: koi bhi do electrons ka bilkul wahi chaar-part address nahi ho sakta — isliye woh spread out hote hain aur isliye har atom alag hota hai!

Active-recall flashcards

Principal quantum number

n

kya determine karta hai?

Shell — main energy level aur average size/nucleus se distance;

n = 1, 2, 3, \dots

l

ki allowed values kya hain?

0

n - 1

tak ke integers

Kisi given

l

ke liye kitni

m_{l}

values hoti hain?

2 l + 1

, ranging

- l

+ l

tak

m_{s}

ki sirf allowed values kya hain?

+ \frac{1}{2}

aur

- \frac{1}{2}

m_{l}

integer kyun hona chahiye?

Φ (ϕ) = e^{i m_{l} ϕ}

ki single-valuedness ke liye

e^{i 2 π m_{l}} = 1

zaroori hai

l

ke terms mein orbital angular momentum ki magnitude?

l (l + 1) ℏ

Shell

n

mein maximum electrons?

2 n^{2}

Kaun sa quantum number sirf magnetic field mein energy affect karta hai?

m_{l}

(magnetic quantum number)

Kya

(3, 2, - 2, + \frac{1}{2})

ek valid set hai?

Haan — yeh ek 3d electron hai (

l = 2 \leq n - 1 = 2

)

f subshell mein kitne electrons fit hote hain?

2 (2 \cdot 3 + 1) = 14

l = 0, 1, 2, 3

ke liye letter codes?

s, p, d, f

Connections

Schrödinger equation for hydrogen atom — teen spatial quantum numbers ki origin
Pauli exclusion principle — koi bhi do electrons chaar numbers share nahi karte
Aufbau principle and electron configuration — $n + l$ se filling order
Zeeman effect — jahan $m_{l}$ finally energy levels split karta hai
Shapes of atomic orbitals (s, p, d) — $l$ ka visual meaning
Hund's rule — unpaired spins maximize karne ke liye $m_{s}$ ka use