Foundations — Spin — intrinsic angular momentum

2.3.17 · D1 · Physics › Modern Physics › Spin — intrinsic angular momentum

Yeh page ek dictionary hai jo tum dekh sakte ho. Parent note Spin ek saath bahut saare symbols throw karta hai ($\vec{S}$, $\hslash$, $m_s$, ${\mu }_B$, commutators…). Yahaan hum unhe ek-ek karke earn karte hain, bilkul kuch nahi se shuru karke, har ek pichle wale par build hota hua.

0 — "Angular momentum" ka matlab hi kya hai

Kisi bhi quantum symbol se pehle, humein spinning motion ke "amount" hone ka ordinary idea chahiye.

Figure dekho: burnt-orange arrow turning ke axis ke saath baitha hai, us direction mein point karta hua jis taraf ek right-handed screw aage badhega. Woh arrow ek vector hai — iski ek length hai (kitna) aur ek direction hai (kaun sa axis). Hum ise orbital motion ke liye $\vec{L}$ likhte hain, aur baad mein spin ke liye $\vec{S}$.

Topic ko yeh kyun chahiye: spin ek angular momentum hai, isliye yeh ek arrow hai. Sab kuch — iski magnitude, iski tilt, iski projection — is arrow ke baare mein koi na koi statement hai.

1 — Components aur $z$-axis: $S_x,S_y,S_z$

Space mein ek arrow ko teen perpendicular rulers par uske shadows se describe kiya ja sakta hai: $x$, $y$, aur $z$ directions.

$z$ kyun aur $x$ nahi? Physically kuch bhi $z$ prefer nahi karta; yeh bas woh direction hai jo hum choose karte hain measure karne ke liye (Stern–Gerlach mein yeh field gradient ki direction hai). Ek baar choose ho jaane ke baad, $S_z$ hai "spin ka up-down amount."

Topic ko yeh kyun chahiye: crucial spin fact — ki arrow $z$ ke saath kabhi bhi fully align nahi ho sakta — ek aisi statement hai ki $S_x$ aur $S_y$ dono zero nahi ho sakte, isliye $|\vec{S}|$ hamesha $S_z$ se bada hota hai. Tum literally yeh components ke bina dekh hi nahi sakte.

2 — Planck's constant $\hslash$: quantum world ka "chunk size"

Nature angular momentum ko koi bhi value lene nahi deti. Yeh chunks mein aata hai, aur ek chunk ki size ek constant se set hoti hai.

$2\pi$ se kyun divide? Ek full turn $2\pi$ radians hota hai. Angular momentum naturally "per radian" measure hota hai, isliye $h/2\pi$ woh version hai jo rotation mein cleanly fit hota hai. Isliye $\hslash$, na ki $h$, har jagah spin mein dikhai deta hai.

Topic ko yeh kyun chahiye: har spin quantity $\hslash$ ke units mein measure hoti hai. $S_z=m_s\hslash$ kehta hai "$S_z$ $\hslash$ ke $m_s$ chunks hai." $\hslash$ ke bina tum yeh nahi bata sakte ki spin kitna bada hai.

3 — Quantum numbers: $s$ aur $m_s$ (plain counting labels)

Kyunki spin chunks mein aata hai, hum unhe count kar sakte hain. Counters pure numbers hain jinhein quantum numbers kaha jaata hai.

Figure dekho: deep-teal $z$-axis par sirf do allowed heights marked hain (arrow ki tip inhi mein se kisi ek par land karni chahiye). Yeh beech mein kahin bhi nahi baithti — yahi "quantized" ka matlab hai. Notice karo ki arrow tilted hai, kabhi flat-up nahi: iski full length $|\vec{S}|$ (plum) dono allowed heights se lambi hai.

Topic ko yeh kyun chahiye: "Stern–Gerlach mein kitne spots?" ka jawab allowed $m_s$ count karke milta hai: $2s+1$ hote hain. Do spots ⇒ $2s+1=2$ ⇒ $s=\frac{1}{2}$. Yahi counting spin ki discovery HAI.

4 — Axis se angle: $\theta$, $\cos$, aur $\arccos$

Arrow ki tilt ek genuine angle hai, aur hum ise trigonometry se measure karte hain — isliye humein ek right triangle chahiye.

Arrow $\vec{S}$ (length $|\vec{S}|$) aur uski $z$-shadow $S_z$ ko drop karo. Yeh ek right triangle banate hain: arrow slanted long side (hypotenuse) hai, aur $S_z$ vertical side hai jo us angle $\theta$ ke saath lagi hai jise hum $z$-axis se measure karte hain.

Cosine kyun aur sine ya tangent nahi? Hum $z$ ke saath wala piece ($\theta$ se adjacent) compare kar rahe hain poore arrow (hypotenuse) se. "Adjacent over hypotenuse" exactly $\cos$ hai. Sine sideways piece compare karta; tangent ko do legs chahiye, leg-aur-hypotenuse nahi — cosine exactly sahi tool hai "arrow ka kitna hissa axis ke upar point karta hai" ke liye.

Topic ko yeh kyun chahiye: parent ka Example 2 electron ki tilt compute karta hai, $\theta =\arccos (1/\sqrt{3})\approx 54.7^{\circ }$. Woh poora result sirf "triangle padho, phir cosine undo karo" hai.

5 — Magnitude versus projection: $\sqrt{s(s+1)}$ kyun, $s$ nahi

Topic mein yeh symbols ka sabse zyada confuse karne wala pair hai, isliye hum inhe clearly alag karte hain.

$s(s+1)$ kyun? Kyunki $|\vec{S}{|}^2=S_x^2+S_y^2+S_z^2$ teeno shadows collect karta hai. $s(s+1)$ ke andar chhupa "+$s$" $S_x,S_y$ ka woh leftover contribution hai jo kabhi vanish nahi ho sakta. Bare $s$ (ya $m_s$) use karna yeh pretend karna hoga ki woh sideways shadows zero hain — yahi classic error hai.

6 — Magnetic moment: $\vec{\mu }$, $e$, $m_e$, $g_s$, aur ${\mu }_B$

Spin physically matter karta hai kyunki ek spinning charge ek tiny bar magnet ki tarah kaam karta hai. Us magnet ki strength magnetic moment hai.

Minus sign kyun? Electron ki charge negative hai, isliye iski magnet arrow uske spin arrow ke opposite direction mein point karti hai. Isliye "spin up" deta hai ${\mu }_z\approx -{\mu }_B$.

Topic ko yeh kyun chahiye: beam split hoti hai kyunki yeh magnet ek aisi field mein force feel karta hai jo position ke saath change hoti hai. Do values of ${\mu }_z$ ⇒ do forces ⇒ do spots.

7 — Deeper machinery ke liye reading key

Parent note mein $[S_x,S_y]=i\hslash S_z$ aur $S_{\pm }=S_x\pm i{S}_y$ bhi likha hai. Tumhe yeh yahaan master karne ki zaroorat nahi — baad ke Deep Dives inhe build karte hain — lekin tumhe symbols padhne aane chahiye:

Topic ko yeh kyun chahiye: yeh woh "first principles" hain jinse $\sqrt{s(s+1)}$ aur $2s+1$ count derived hote hain bjaaye memorise kiye jaane ke.

Prerequisite map

Equipment checklist

Khud ko test karo — main Spin note padhne se pehle har ek ka jawab dene mein comfortable hona chahiye.

Connections

• Parent: Spin — woh topic jiske liye yeh page tumhe prepare karta hai.
• Orbital Angular Momentum — same vector/component machinery, integer $m$.
• Quantum Numbers — $s$ aur $m_s$ $n,\ell ,m_{\ell }$ mein kahan baithte hain.
• Stern-Gerlach Experiment — jahan $2s+1$ count visible spots ban jaata hai.
• Bohr Magneton — ${\mu }_B$ unit yahaan build hoti hai.
• Dirac Equation — explain karta hai ki $g_s\approx 2$ kyun hai.
• Zeeman Effect · Fine Structure · Pauli Exclusion Principle — downstream uses.