Foundations — Quantum numbers n, l, mₗ, mₛ

2.3.13 · D1 · Physics › Modern Physics › Quantum numbers n, l, mₗ, mₛ

Parent note padhne se pehle, tumhe uski har vocabulary piece apni karni hogi. Yeh page har ek symbol ko bilkul zero se build karta hai: uska kya matlab hai plain words mein, uske liye kaunsi picture kaam aati hai, aur kyun yeh topic uske bina kaam nahi kar sakta. Upar se neeche padho — har item apne pehle waale par depend karta hai, aur koi bhi quantum number use hone se pehle derive kiya jaata hai.

1. Angle, radius, aur teen coordinates $(r,\theta ,\varphi )$

Kisi bhi physics se pehle, hume ek tarika chahiye yeh batane ka ki nucleus ke aaspaas ek point kahan hai.

Aam tarika hai $(x,y,z)$: east chalo, north chalo, upar chalo. Lekin ek atom gol hota hai — nucleus ki pulling force kisi bhi direction mein ek given distance par same hoti hai. Isliye hum aise coordinates use karte hain jo uss golai se match karti hain: spherical coordinates.

Figure dekho: $r$ amber arrow ki length hai, $\theta$ yeh hai ki woh arrow seedha upar se kitna jhuka hai, aur $\varphi$ yeh hai ki tum equator ke around $+x$-axis se shuru karke kitna sweep kar chuke ho. Space ka har point exactly ek triple $(r,\theta ,\varphi )$ se describe hota hai.

Yeh topic ko kyun chahiye: electron wave ko $\psi (r,\theta ,\varphi )$ likha jaata hai (§3 mein milenge $\psi$ se). Baad mein, teen "shape counts" mein se har ek in teeno directions mein se ek se aayega — ek $r$ se, ek $\theta$ se, ek $\varphi$ se.

2. $\pi$ aur radians ka matlab

$\varphi$ $0$ se $2\pi$ tak gaya, $0$ se $360°$ tak nahi. Kyun?

3. Wave $\psi$ (psi)

Yeh topic ko kyun chahiye: "electron ek standing wave hai" ka poora idea wohi statement hai ki electron $\psi$ se describe hota hai. Quantum numbers allowed $\psi$ shapes ke labels hain.

4. Standing waves aur "poore number of bumps fit karna"

Yahi integers kahan se aate hain, iska dil hai.

Figure dekho: string par tum 1 bump, 2 bumps, 3 bumps fit kar sakte ho — lekin kabhi $2.5$ bumps nahi, kyunki loose end line up nahi hoga. Bumps ki ginti poora number honi forced hai. Yahi poora number har quantum number ka ancestor hai.

5. Complex numbers aur $e^{i\varphi }$ — aur $m_l$ ka janam

$\varphi$-wave ko $\Phi (\varphi )=e^{i{m}_l\varphi }$ likha jaata hai. Woh $i$ aur $e$ logon ko dara deta hai. Yahan sab kuch hai jo tumhe chahiye.

Ab hum apna pehla quantum number derive karte hain. Electron ki $\varphi$-direction ko ek ring mein mod do (§1 ka $\varphi$ circle, $+x$ se shuru). Ek full loop chalte waqt tum $\varphi$ se $\varphi +2\pi$ jaate ho, aur wave ko same value par waapas aana hoga — warna usmein ek rip hogi: $\Phi (\varphi +2\pi )=\Phi (\varphi )\Rightarrow e^{i{m}_l(\varphi +2\pi )}=e^{i{m}_l\varphi }\Rightarrow e^{i{m}_l\cdot 2\pi }=1.$

Yeh kya kehta hai: circle ke around $m_l$ baar jaane par tumhe exactly waapas start par aana chahiye. Yeh tabhi true hota hai jab $m_l$ ek whole number ho.

Yeh topic ko kyun chahiye: $m_l$ pehla integer hai jo ek match-yourself condition se "gir ke aata hai." Uska sign ek real physical cheez hai — yeh batata hai ki electron fog kis direction mein circulate ho raha hai.

6. $\theta$-wave aur $l$ ka janam

$\varphi$-direction ne $m_l$ diya. Ab tilt direction $\theta$ ek doosra whole number deta hai.

Tilt-wave $\Theta (\theta )$ ko Schrödinger equation ka polar piece manna hoga (jise Legendre's equation kehte hain). Tumhe ise solve karne ki zaroorat nahi — tumhe ek fact chahiye.

Jab tum "dono poles par finite" enforce karte ho, toh maths (Legendre's equation) sirf tabhi solutions allow karta hai jab ek certain constant $l(l+1)$ ke barabar ho aur $l$ ek whole number $0,1,2,\dots$ ho — aur sirf jab $|m_l|\le l$ ho. Koi bhi doosri value tilt-wave ko ek pole par infinity tak shoot kar deti hai, jo unphysical hai (fog kisi ek point par infinitely thick nahi ho sakta).

$|m_l|\le l$ kyun sense karta hai: tum equator ke around utne sharply wrap nahi kar sakte jitna overall pattern wrinkled hai. Ek bahut "calm" pattern (chhota $l$) ke paas simply fast horizontal wrap (bada $|m_l|$) ke liye jagah nahi hai. Figure dekho: jaise $l$ badhta hai, negative aur positive dono sides par zyada $m_l$ slots khulte jaate hain.

7. $r$-wave aur $n$ ka janam

Do ho gaye, ek baki hai. Radial direction $r$ wave-equation trio ka aakhiri number deta hai.

Jab tum "wave $r\to \infty$ par khatam ho jaaye" enforce karte ho, radial equation ke paas sirf whole-number values $n=1,2,3,\dots$ ke liye well-behaved solutions hoti hain, aur sirf jab $l\le n-1$ ho. Yahi condition allowed energies bhi freeze karti hai (yahan se parent note ka $E_n=-13.6\text{eV}/n^2$ aata hai).

Yeh topic ko kyun chahiye: $n$ master label hai — yeh energy aur size set karta hai, aur yeh cap karta hai ki fog kitni wrinkled ($l$) ho sakti hai. Isse produce hone wali energies ke liye Hydrogen Atom Energy Levels dekho.

8. Vectors, arrow $\vec{L}$, aur constant $\hslash$

Hamare paas ab $n,l,m_l$ hain. $l$ aur $m_l$ ko real physics se connect karne ke liye ek aur picture chahiye: angular momentum ka arrow.

Ab do integers apna physical kaam kamaate hain:

• $l$ arrow ki length set karta hai: $|\vec{L}|=\sqrt{l(l+1)}\hslash$.
• $m_l$ arrow ka shadow set karta hai: $L_z=m_l\hslash$.

Figure dekho: cyan arrow $\vec{L}$ hai, aur vertical $z$-axis par uska shadow $L_z$ hai. Shadow hamesha arrow se chhota hota hai jab tak arrow seedha upar point na kare — yahi exactly woh reason hai jis par parent note insist karta hai ki $L_z<|\vec{L}|$ (space quantization). Aur kyunki $m_l$ negative ho sakta hai, shadow neeche ($L_z<0$) bhi point kar sakta hai, sirf upar nahi.

Yeh topic ko kyun chahiye: ek $l=0$ state mein $|\vec{L}|=0$ hota hai — ek perfectly round, non-swirling fog. Poori arrow ki kahani ke liye Angular Momentum in Quantum Mechanics dekho.

9. Spin: numbers $s$ aur $m_s$

Teen numbers $n,l,m_l$ sab wave equation se aaye. Ek chautha label hai jo us equation se nahi aata — woh humpar experiment ne force kiya.

Jaise tilt-arrow $\vec{L}$ ka shadow $L_z=m_l\hslash$ tha aur $m_l$ whole steps mein $-l$ se $+l$ tak jaata tha, spin-arrow $\vec{S}$ ka shadow $S_z=m_s\hslash$ hota hai jiska label $m_s$ whole steps mein $-s$ se $+s$ tak jaata hai.

Exactly do values kyun? Kyunki $s=\frac{1}{2}$ hai, $-s$ se $+s$ tak ki ladder mein sirf do rungs hain $\pm \frac{1}{2}$. Yahi exactly Stern-Gerlach Experiment ne dikhaya: atoms ki ek beam exactly do spots mein split hui — ek two-valued built-in magnet ka direct fingerprint.

10. Schrödinger equation (woh machine jo yeh sab banati hai)

Tumhe ise solve karne ki zaroorat nahi (yeh Schrödinger Equation ka kaam hai). Tumhe sirf itna chahiye: yeh sieve hai; pehle teen quantum numbers jo bhi isse pass karta hai uske labels hain.

Prerequisite map

Equipment checklist

Right side cover karo aur khud ko test karo. Agar tum yeh sab answer kar sako, toh parent note 2.3.13 smoothly padhega.