Quantum numbers n, l, mₗ, mₛ: Question bank — Hinglish deep dive

2.3.13 · D5 · Physics › Modern Physics › Quantum numbers n, l, mₗ, mₛ

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Definition Is page par har jagah use hone wale symbols

$ℏ$ (padho "h-bar") the reduced Planck constant hai, $ℏ = h /2 π$ . Ye ek fixed constant of nature hai (ek specific measured number, koi unit nahi jo humne choose ki) jo quantum mechanics mein angular momentum ka scale set karta hai — har angular momentum kisi plain number times $ℏ$ nikalta hai. Toh $ℏ$ "ek metre" jaisa nahi hai; ye zyada fixed speed of light $c$ jaisa hai: ek built-in physical constant jiske against sab kuch measure hota hai.
$L$ electron ka orbital angular-momentum vector hai — ek arrow jiska length bolta hai kitni rotational motion orbital carry karta hai aur jiska direction bolta hai "rotation axis" kis taraf point karta hai.
$∣ L ∣$ us arrow ki length hai (ek single positive number), jabki $L_{z}$ vertical $z$ -axis par uska shadow hai (uska projection — positive, zero, ya negative ho sakta hai).
$θ$ (theta) aur $ϕ$ (phi) wo do angles hain jo sphere par kahin bhi point karne ke liye use hote hain: $θ$ polar angle hai jo upar ( $z$ ) axis se niche naap-a jaata hai (latitude ki tarah, north pole par $0$ , south par $π$ ), aur $ϕ$ azimuthal angle hai jo equator ke around naapa jaata hai (longitude ki tarah, $0$ se $2 π$ tak). Niche figure mein sphere dekho.
$m_{s}$ spin magnetic quantum number hai, electron ke intrinsic (built-in) spin direction ka label. Ye sirf do values leta hai, $+ \frac{1}{2}$ aur $- \frac{1}{2}$ ("spin up" aur "spin down"), aur $n, l, m_{l}$ ke unlike ye space mein wave ki shape se nahi aata — ise experiment (Stern–Gerlach) ne discover kiya tha.

True or false — justify karo

Niche har item ek statement hai. True ya false decide karo, aur — ye hi poori baat hai — kyun.

TRUE or FALSE: Quantum numbers wo labels hain jo physicists ne convention se choose kiye, jaise particles ka naam rakhna.

False. Ye Schrödinger's equation solve karte waqt boundary conditions (single-valuedness, non-divergence, infinity par decay) se nikalta hai; integers mathematics se forced hain, choice se assign nahi kiye gaye.

TRUE or FALSE:

l = 0

ke liye electron ka orbital angular momentum zero hota hai.

True.

∣ L ∣ = l (l + 1) ℏ = 0 ℏ = 0

; ek s-electron ka cloud spherically symmetric hota hai aur "circulate" nahi karta.

TRUE or FALSE: Orbital angular momentum ki magnitude

l ℏ

ke barabar ho sakti hai.

False. Ye

l (l + 1) ℏ

ke barabar hoti hai, jo har

l \geq 1

ke liye

l ℏ

se strictly zyada hai (jaise

l = 1

deta hai

2 ℏ > 1ℏ

);

l ℏ

purana Bohr ka guess tha.

TRUE or FALSE:

m_{l}

zero ho sakta hai jab bhi

l

bada ho.

True.

m_{l}

run karta hai

- l, \dots, 0, \dots, + l

, toh

0

hamesha list mein hota hai; iska matlab hai

L

x y

-plane mein hai aur koi

z

-projection nahi hai.

TRUE or FALSE:

n = 4

wala shell zyada se zyada 32 electrons rakh sakta hai.

True.

2 n^{2} = 2 (4)^{2} = 32

, agar

n = 4

ke liye allowed sabhi

(l, m_{l}, m_{s})

combinations gine jayein.

TRUE or FALSE: Ek hi atom mein do electrons

n

l

, aur

m_{l}

share kar sakte hain agar unke spins alag hon.

True. Pauli sirf charon share karne ko forbid karta hai;

m_{s} = + \frac{1}{2}

aur

m_{s} = - \frac{1}{2}

full sets ko distinct banate hain, toh ek orbital exactly do electrons rakhta hai.

TRUE or FALSE: Spin quantum number

m_{s}

Schrödinger equation se nikalta hai.

False. Basic (non-relativistic) Schrödinger equation sirf

n, l, m_{l}

deta hai; spin ko Stern-Gerlach Experiment ne forced kiya aur baad mein Dirac ki relativistic equation ne derive kiya.

TRUE or FALSE:

n

badhane par hamesha possible

l

values ki sankhya barhti hai.

True.

l

range karta hai

0

n - 1

tak, toh exactly

n

possible

l

values hain; bada

n

matlab ek aur subshell type available ho jaata hai.

TRUE or FALSE: Vector

L

exactly

z

-axis ke saath point kar sakta hai.

False. Uske liye

L_{z} = ∣ L ∣

chahiye hoga, yaani

m_{l} ℏ = l (l + 1) ℏ

, lekin sabse bada allowed

m_{l}

hai

l

aur

l < l (l + 1)

for

l \geq 1

; eigenvalues allow hi nahi karte, toh

L

hamesha tilted rehta hai (space quantization — cones figure dekho).

Error dhundo

Har line mein ek flaw wala claim hai. Reveal karo flaw ka naam aur usse fix karo.

"For

n = 2

, the allowed

l

values are

1

and

2

Wrong upper limit —

l

rukta hai

n - 1 = 1

par, toh sirf

l = 0, 1

hain;

l = 2

ke liye

n \geq 3

chahiye hoga.

m_{l}

orbitals count karta hai, isliye ye run karta hai

0, 1, \dots, l

m_{l}

ek signed projection hai, count nahi; ye run karta hai

- l

+ l

tak, jo negatives samet

2 l + 1

values deta hai.

n = 1, l = 1

wale electron mein ek 1p orbital hai."

1p exist hi nahi karta —

n = 1

ke liye

l \leq n - 1 = 0

, toh ek hi

n = 1

subshell hai 1s (

l = 0

"Spin ka matlab hai electron physically ek axis ke around rotate karta hai."

Spin intrinsic angular momentum hai jisme koi classical rotation nahi; ek rotating charged sphere ki surface ko light speed se zyada jaana padta, toh picture sirf ek mnemonic hai.

"Kyunki

L_{z} = m_{l} ℏ

aur

m_{l}

bada ho sakta hai,

L_{z}

∣ L ∣

se exceed kar sakta hai."

Impossible — sabse bada allowed value hai

m_{l} = l

, aur

l ℏ < l (l + 1) ℏ

, toh

∣ L_{z} ∣

(shadow) hamesha strictly

∣ L ∣

(full arrow) se kam hota hai.

"Set

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

ek valid state hai."

Invalid —

∣ m_{l} ∣ \leq l = 1

, lekin

m_{l} = - 2

us cap ko violate karta hai; allowed

m_{l}

sirf

- 1, 0, + 1

hain.

m_{s} = 0

ek electron ke liye allowed hai jiska koi preferred spin direction nahi."

Ek spin-

\frac{1}{2}

particle ke liye sirf

m_{s} = \pm \frac{1}{2}

exist karte hain; koi

m_{s} = 0

state nahi hoti — spin projection hamesha half-integer hota hai.

"Energy charon quantum numbers par depend karti hai."

Pure hydrogen atom ke liye, energy sirf

n

par depend karti hai (

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

eV);

l, m_{l}, m_{s}

isse unchanged chhodte hain (degeneracy), jo sirf finer effects se toot-ti hai.

Why questions

WHY single-valuedness condition

Φ (ϕ + 2 π) = Φ (ϕ)

m_{l}

ko integer hone par kyun force karti hai?

Kyunki

Φ = e^{i m_{l} ϕ}

ko azimuthal angle

ϕ

ke around poora loop karne ke baad khud par wapas aana chahiye, jiske liye

e^{i m_{l} 2 π} = 1

chahiye, jo tab hi hota hai jab

m_{l}

ek whole number ho — non-integer wave ko same physical point par multi-valued bana dega.

WHY

l

m_{l}

\pm l

par kyun cap karta hai instead of koi bhi value allow karne ke?

Angle

θ

mein polar (Legendre) solutions poles

θ = 0, π

par tab hi finite rehte hain jab

∣ m_{l} ∣ \leq l

; zyada bada

∣ m_{l} ∣

wavefunction ko wahan blow up kara deta hai, jo unphysical hai.

WHY sabse bada

L_{z}

kabhi

∣ L ∣

ke barabar kyun nahi ho sakta?

Eigenvalues fix karte hain

L_{z} = m_{l} ℏ

maximum

m_{l} = l

ke saath, lekin

∣ L ∣ = l (l + 1) ℏ

, aur har

l \geq 1

ke liye

l < l (l + 1)

hai; toh sabse tall allowed shadow

l ℏ

bhi arrow ki asli length se chhota rehta hai — vector tilted rehna padta hai.

WHY ek s-orbital (

l = 0

) ki shape spherical kyun hoti hai?

Zero orbital angular momentum ka matlab hai koi preferred "circulation" direction nahi, toh probability cloud ka koi angular pattern nahi aur har direction mein same dikhta hai.

WHY shell

n

mein electrons ki maximum sankhya

2 n^{2}

kyun hai aur sirf

2 n

nahi?

Har subshell

l

mein

2 (2 l + 1)

electrons hote hain (

2 l + 1

orbitals times 2 spins), toh tum sum karte ho

\sum_{l = 0}^{n - 1} 2 (2 l + 1) = 4 \sum_{l = 0}^{n - 1} l + 2 n = 4 \cdot \frac{( n - 1 ) n}{2} + 2 n = 2 n^{2} - 2 n + 2 n = 2 n^{2}

; arithmetic-series sum

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

hi isko collapse karta hai.

WHY Stern–Gerlach experiment ne ek fourth quantum number demand kyun kiya?

Silver atoms ki ek beam exactly do spots mein split hui, ek two-valued magnetic degree of freedom reveal karti hui jo

n, l, m_{l}

se account nahi hoti thi — ye intrinsic spin hai,

m_{s} = \pm \frac{1}{2}

se label kiya gaya.

WHY

l

n

tak pahunchne ki bajaye

n

se ek chhota kyun rukta hai?

Radial wave ka ek valid bound-state form hona chahiye jo infinity par decay kare, aur radial structure ka "budget" require karta hai

l \leq n - 1

;

l = n

set karne par koi admissible square-integrable radial solution nahi bachta.

Edge cases

WHAT

n = 1

hone par quantum-number ki saari possibilities kya hain?

Sirf

l = 0

m_{l} = 0

, aur

m_{s} = \pm \frac{1}{2}

— exactly do states (1s²); yahan koi p, d, ya f subshells exist nahi kar sakti.

WHAT kisi bhi s-orbital mein electron ke liye

∣ L ∣

kya hai,

n

chahe kuch bhi ho?

Exactly zero, kyunki

l = 0

deta hai

∣ L ∣ = 0 (0 + 1) ℏ = 0

;

n

ki value energy aur size change karti hai lekin ye angular momentum nahi.

WHAT

l \to 0

hone par allowed

m_{l}

values ki sankhya ka kya hoga?

Ye shrink hokar

2 (0) + 1 = 1

ho jaati hai; ek single

m_{l} = 0

state hoti hai, matlab ek s-subshell mein sirf ek orbital hota hai (do electrons spin ke saath).

WHAT wo sabse chhota

n

kya hai jiske liye ek d-electron (

l = 2

) exist kar sakta hai?

n = 3

, kyunki

l = 2

ke liye

l \leq n - 1

chahiye, yaani

n \geq 3

; koi 1d ya 2d nahi hota.

WHAT

m_{l} = 0

ek nonzero-

l

orbital ke liye geometrically kya matlab rakhta hai?

Angular-momentum vector ka koi

z

-component nahi hai (

L_{z} = 0

), toh

L

completely

z

ke perpendicular plane mein hai jabki uski full magnitude

l (l + 1) ℏ

bani rehti hai.

WHAT

n = 3

ke liye distinct

(n, l, m_{l}, m_{s})

states ki total sankhya kya hai?

2 n^{2} = 2 (3)^{2} = 18

states — 3s (2), 3p (6), aur 3d (10) capacities milake.

WHAT

l = n - 1

(maximum

l

) ke case mein kya special hai?

Orbital mein us shell ki sabse zyada angular structure hoti hai aur sabse kam radial nodes hote hain; ye Bohr analogy mein "sabse circular" orbit hai, classical orbit ke sabse kareeb.