Quantum numbers n, l, mₗ, mₛ — Hinglish notes

2.3.13 · Physics › Modern Physics

WHAT are the four quantum numbers?

Definition Char quantum numbers

Symbol	Name	Allowed values	Physically controls
$n$	Principal quantum number	$1, 2, 3, \dots$	energy aur orbital ki overall size
$l$	Azimuthal (orbital angular momentum)	$0, 1, 2, \dots, (n - 1)$	orbital ki shape, angular momentum ki magnitude
$m_{l}$	Magnetic quantum number	$- l, \dots, 0, \dots, + l$	space mein orbital ki orientation ( $L$ ka $z$ -component)
$m_{s}$	Spin magnetic quantum number	$+ \frac{1}{2}, - \frac{1}{2}$	electron ki intrinsic spin direction

Pehle teen ( $n, l, m_{l}$ ) Schrödinger's equation ko spherical coordinates $(r, θ, ϕ)$ mein 3D problem ke liye solve karne se aate hain. Chautha ( $m_{s}$ ) extra hai — yeh basic Schrödinger equation mein appear nahi karta; yeh experiment (Stern–Gerlach) ne hum par thopa aur baad mein Dirac's relativistic equation ne properly derive kiya.

WHY do quantum numbers exist? (Derivation from scratch)

Step 1 — Wavefunction ko separate karo. Hum 3D wavefunction ko ek product ke roop mein likhte hain: $ψ (r, θ, ϕ) = R (r) Θ (θ) Φ (ϕ)$ Yeh step kyun? Coulomb potential $V = - \frac{k e ^{2}}{r}$ sirf $r$ par depend karta hai, isliye equation saaf taur par ek radial part aur ek angular part mein separate ho jaati hai. Har part apna quantum number deta hai.

Step 2 — $ϕ$ equation se $m_{l}$ milta hai. Azimuthal part iska palan karta hai: $\frac{d ^{2} Φ}{d ϕ ^{2}} = - m_{l}^{2} Φ \Rightarrow Φ (ϕ) = e^{i m_{l} ϕ} .$ Yeh step kyun? $Φ$ ek single-valued physical wave hone ke liye humein chahiye $Φ (ϕ + 2 π) = Φ (ϕ)$ , yani $e^{i m_{l} 2 π} = 1$ . Yeh tabhi sach hai jab $m_{l}$ ek integer ho. Yahin se integers aate hain — ek self-consistency (boundary) condition se.

Step 3 — $θ$ equation se $l$ milta hai. Polar equation (Legendre's equation) solve karne par finite, non-blowing-up solutions tabhi milte hain jab separation constant $l (l + 1)$ ke barabar ho, jahan $l = 0, 1, 2, \dots$ , aur tabhi jab $∣ m_{l} ∣ \leq l$ ho. Yeh step kyun? Agar $l$ integer na hota ya agar $∣ m_{l} ∣ > l$ hota, toh solution poles ( $θ = 0, π$ ) par diverge ho jaata — jo unphysical hai. Isliye $l$ , $m_{l}$ ko cap karta hai.

Step 4 — $r$ equation se $n$ milta hai. Radial equation ke well-behaved (square-integrable) solutions tabhi milte hain jab $n = 1, 2, 3, \dots$ ho aur $l \leq n - 1$ bhi ho. Yeh step kyun? Boundary condition ab yeh hai ki "wave ko $r \to \infty$ par khatam ho jaana chahiye." Yahi bound-state condition energy ko quantize karti hai.

$L_{z}$ quantized kyun hai lekin kabhi $∣ L ∣$ ke barabar kyun nahi? Kyunki $m_{l} \leq l < l (l + 1)$ . Vector $L$ kabhi poori tarah $z$ ki taraf point nahi kar sakta — agar karta, toh $L_{x} = L_{y} = 0$ exactly ho jaate, jo angular momentum ke liye uncertainty principle ka violation hoga. Yahi "space quantization" hai: $L$ sirf discrete tilt angles par allowed hai.

HOW to use them — counting states

Diye gaye $l$ ke liye $m_{l}$ ki allowed values ki sankhya $2 l + 1$ hai ( $- l$ se $+ l$ tak). Spin ( $\times 2$ ) include karne par, ek shell mein electrons ki sankhya hoti hai: $\sum_{l = 0}^{n - 1} 2 (2 l + 1) = 2 n^{2} .$ Kyun? Har distinct $(n, l, m_{l}, m_{s})$ ek quantum state hai, aur (Pauli ke anusaar) koi bhi do electrons charon ek saath share nahi karte.

Common mistake Steel-manned common mistakes

Mistake A: " $m_{l}$ , $0$ se $l$ tak jaata hai." Yeh sahi kyun lagta hai: counting aksar 0 se shuru hoti hai, aur $m_{l}$ ek count jaisi lagti hai. Fix: $m_{l}$ ek axis par ek vector ka projection hai, isliye yeh negative ho sakta hai: $m_{l} = - l, \dots, + l$ , jo $2 l + 1$ values deta hai.

Mistake B: "Angular momentum $l ℏ$ hai (Bohr ka purana rule)." Yeh sahi kyun lagta hai: Bohr's model ne $L = n ℏ$ diya tha, aur yeh simpler hai. Fix: Schrödinger deta hai $∣ L ∣ = l (l + 1) ℏ$ , naa ki $l ℏ$ . $l = 0$ ke liye, $∣ L ∣ = 0$ (bilkul bhi orbital angular momentum nahi — ek s-electron ka ek spherical, non-rotating cloud hota hai).

Mistake C: "Spin ka matlab hai ki electron ek top ki tarah physically ghoomta hai." Yeh sahi kyun lagta hai: naam aur $ℏ$ units rotation suggest karte hain. Fix: Spin ek intrinsic property hai jismein koi classical rotation nahi; literally ghoomti hui surface light speed se zyada ho jaayegi. $m_{s}$ ke sirf do values hain, $\pm \frac{1}{2}$ , aur koi bhi "ghoomne ki" spatial direction nahi hai.

Mistake D: " $n = 1$ mein ek p-orbital ho sakta hai." Yeh sahi kyun lagta hai: har doosre shell mein p, d… hote hain. Fix: $l \leq n - 1$ , isliye $n = 1$ sirf $l = 0$ (1s) allow karta hai. 1p exist hi nahi karta.

Recall Feynman: 12-saal ke bacche ko explain karo

Socho electron ek chhoti drum skin jaisi hai jo nucleus ke around wrapped hai. Ek drum sirf kuch certain notes baja sakta hai — aap beech wala note nahi nikal sakte. $n$ batata hai drum kitna bada hai (bada drum = low note = nucleus se door). $l$ vibration ka pattern batata hai (shant aur round, ya stripes ke saath). $m_{l}$ batata hai woh striped pattern kis taraf ghuma hua hai. Aur $m_{s}$ electron mein ek tiny built-in magnet hai jo sirf "upar" ya "neeche" point kar sakta hai. Ek hi atom mein koi bhi do electrons ko exactly same char labels rakhne ki permission nahi hai — jaise assigned seats jinka unique seat number hota hai.

Recall Active recall — answers chhupa lo

Kaunsi boundary condition $m_{l}$ ko integer hone par majboor karti hai?
$∣ L ∣ \neq = l ℏ$ kyun hai?
Shell $n$ mein maximum electrons kitne?

Principal quantum number

n

physically kya control karta hai?

Orbital ki energy aur overall size; hydrogen ke liye

E_{n} = - 13.6/ n^{2}

eV.

Diye gaye

n

ke liye

l

ki allowed values kya hain?

l = 0, 1, 2, \dots, (n - 1)

Diye gaye

l

ke liye

m_{l}

ki allowed values kya hain?

m_{l} = - l, - l + 1, \dots, 0, \dots, + l

(yaani

2 l + 1

values).

m_{s}

ki sirf allowed values kya hain?

+ \frac{1}{2}

aur

- \frac{1}{2}

n, l, m_{l}

mathematically kahan se aate hain?

Schrödinger equation ko spherical coordinates mein solve karte waqt boundary/single-valued conditions se (radial → n, polar → l, azimuthal → m_l).

Kaunsi boundary condition

m_{l}

ko quantize karti hai?

Single-valuedness:

Φ (ϕ + 2 π) = Φ (ϕ)

, toh

e^{i m_{l} 2 π} = 1

, jo

m_{l}

ko integer hone par majboor karta hai.

Orbital angular momentum ki magnitude ka formula?

∣ L ∣ = l (l + 1) ℏ

Orbital angular momentum ke

z

-component ka formula?

L_{z} = m_{l} ℏ

L

kabhi poori tarah

z

-axis ki taraf point kyun nahi kar sakta?

Kyunki

m_{l} \leq l < l (l + 1)

, isliye

L_{z} < ∣ L ∣

; poori tarah point karna

L_{x} = L_{y} = 0

fix kar deta, jo angular-momentum uncertainty ka violation hoga (space quantization).

Shell

n

mein kitne electrons aa sakte hain aur kyun?

2 n^{2}

\sum_{l = 0}^{n - 1} 2 (2 l + 1)

se, jahan factor 2 spin ke liye hai.

Kya

(n, l, m_{l}, m_{s}) = (2, 2, 0, \frac{1}{2})

allowed hai?

Nahi —

l

\leq n - 1 = 1

hona chahiye, isliye

l = 2

forbidden hai.

S-electron (

l = 0

) ke liye

∣ L ∣

kya hai?

Zero, kyunki

0 (0 + 1) ℏ = 0

WHAT are the four quantum numbers?

WHY do quantum numbers exist? (Derivation from scratch)

HOW to use them — counting states

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