Spin — intrinsic angular momentum — Hinglish notes

2.3.17 · Physics › Modern Physics

Spin kyun exist karta hai? (experimental forcing)

Spin kya hai? (definition)

Electron ke liye $s = \frac{1}{2}$ , to $m_{s} = \pm \frac{1}{2}$ → exactly do states ("spin up" $↑$ , "spin down" $↓$ ). Isi wajah se Stern–Gerlach ne do spots diye.

Numbers kaise nikalte hain (derive karo, memorise mat karo)

Spin wahi algebra follow karta hai jo koi bhi angular momentum karta hai. Hum ise pehla principle mante hain: $[S_{x}, S_{y}] = i ℏ S_{z} (and cyclic) .$

Yeh kyun matter karta hai: yeh commutators poori foundation hain. Neeche sab kuch inhi se nikalta hai, bilkul waise jaise orbital $L$ ke liye.

Algebra se, operator $S^{2}$ har component ke saath commute karta hai, to ek state ki definite total magnitude ho sakti hai. Uska eigenvalue hai: $S^{2} ψ = ℏ^{2} s (s + 1) ψ .$ $s (s + 1)$ kyun, $s^{2}$ kyun nahi? Kyunki $S^{2} = S_{x}^{2} + S_{y}^{2} + S_{z}^{2}$ ; "extra" $+ s$ components $S_{x}, S_{y}$ ka contribution hai jo kabhi simultaneously zero nahi ho sakte (uncertainty). Isliye: $∣ S ∣ = ℏ s (s + 1) .$

Step 2 — projection

$S_{z}$ , $ℏ$ ke steps mein quantized hai: $S_{z} = m_{s} ℏ, m_{s} = - s, - s + 1, \dots, + s .$ Yeh $- s$ se $+ s$ tak kyun jaate hain? Ladder operators $S_{\pm} = S_{x} \pm i S_{y}$ , $m_{s}$ ko 1 se raise/lower karte hain; ladder top aur bottom par terminate karni chahiye, jo range ko symmetric aur $2 s$ ko integer banene pe majboor karti hai. $s = \frac{1}{2}$ ke liye sirf allowed values $m_{s} = \pm \frac{1}{2}$ hain.

Step 3 — magnetic moment

Angular momentum wale ek charged particle ka magnetic moment hota hai. Spin ke liye: $μ_{s} = - g_{s} \frac{e}{2 m _{e}} S$ Extra factor $g_{s}$ kyun? Classically ek current loop $μ = \frac{e}{2 m} L$ deta hai (yeh " $g = 1$ " wala guess hai). Experiment aur Dirac ki relativistic theory dikhate hain ki electron ka spin do guna zyada magnetic hai: $g_{s} \approx 2$ . Yeh anomalous " $g = 2$ " ek gehri, purely quantum result hai.

Worked examples

Forecast-then-Verify

Common mistakes (steel-manned)

Flashcards

Spin kya hai?

Ek particle ka intrinsic, fixed angular momentum, spatial motion se unrelated, possibly half-integer quantum number ke saath quantized.

Electron ka spin quantum number

s = \frac{1}{2}

Electron ke liye allowed

m_{s}

values

+ \frac{1}{2}

aur

- \frac{1}{2}

(do states).

Electron spin ki magnitude

∣ S ∣ = ℏ s (s + 1) = \frac{3}{2} ℏ

Stern–Gerlach ne do spots kyun diye?

Kyunki

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

→ magnetic moment ki exactly do orientations hain.

Spin

s

ke liye Stern–Gerlach spots ki sankhya

2 s + 1

Electron ka spin g-factor

g_{s} \approx 2

(anomalous; 1 nahi).

Bohr magneton expression

μ_{B} = \frac{e ℏ}{2 m _{e}}

, electron ke spin

z

-moment ki magnitude.

S_{z}

aur

∣ S ∣

mein fark

S_{z} = m_{s} ℏ

projection hai;

∣ S ∣ = ℏ s (s + 1)

full magnitude hai (bada).

Electron spin ka z-axis se angle

θ = arccos \frac{1}{3} \approx 54. 7^{\circ}

Field gradient mein spin par force

F_{z} = μ_{z} \partial B_{z} / \partial z

, jahan

μ_{z} = \mp μ_{B}

Recall Feynman: 12-saal ke bachche ko explain karo

Socho har electron factory se ek chhota sa built-in "twirl" lekar aata hai. Tum ise zyada ya kam nahi kar sakte — yeh hamesha utna hi twirl karta hai. Jab tum ise magnet ke paas rakhte ho, to yeh apna twirl sirf do tareekon se point kar sakta hai: thoda upar, ya thoda neeche. Isliye atoms ki ek beam neatly do mein split ho jaati hai — smear nahi, bas do. Twirl koi real spinning ball nahi hai; yeh quantum world ka ek rule hai. Hum is built-in twirl ko spin kehte hain.

Connections

Stern-Gerlach Experiment — woh experiment jisne spin reveal kiya.
Orbital Angular Momentum — same algebra, sirf integer quantum numbers.
Quantum Numbers — $n, ℓ, m_{ℓ}, m_{s}$ aur full state label.
Pauli Exclusion Principle — spin- $\frac{1}{2}$ (fermions) par depend karta hai.
Zeeman Effect & Fine Structure — $U = - μ \cdot B$ se energy splitting.
Bohr Magneton — magnetic moment ki natural unit.
Dirac Equation — $g_{s} = 2$ ko first principles se explain karta hai.

Spin — intrinsic angular momentum