Hydrogen energy levels Eₙ = −13.6 - n² eV: Visual walkthrough

2.3.14 · D2 · Physics › Modern Physics › Hydrogen energy levels Eₙ = −13.6 - n² eV

Kirdaar (pehle define karte hain, phir use karte hain)

Picture dekho: proton (magenta dot) still baitha hai, electron (violet dot) usse $r$ distance par circle kar raha hai, orange arrow ki direction mein speed $v$ se chal raha hai. Yahi poora physical system hai. Ab hum poochte hain: electron ko us circle mein kya rokta hai, aur uski kitni energy hai?

Step 1 — Electron ko kya kheenchta hai? (Coulomb's law)

KYA. Opposite charges attract karte hain. Proton ( $+ e$ ), electron ( $- e$ ) ko seedha centre ki taraf kheenchta hai ek force se jise hum $F_{pull}$ kehte hain.

YEH TOOL KYUN. Hume jaanna hai ki pull kitni strong hai, kyunki woh strength decide karti hai ki electron kitni fast aur kitni door orbit kar sakta hai. Coulomb's law exactly woh rule hai jo batata hai "do charges ke beech electric pull kitni strong hai jab woh $r$ door hain?"

PICTURE. Force electron se seedha proton ki taraf point karti hai — hamesha andar, radius ke along.

Step 2 — Woh pull circle kyun banata hai? (Centripetal force)

KYA. Jo bhi cheez circle mein move karti hai woh continuously centre ki taraf mod rahi hoti hai. Mass $m$ ko speed $v$ se radius $r$ ke circle mein bend karne ke liye jitni inward force chahiye woh hoti hai centripetal force, $F_{turn} = \frac{m v ^{2}}{r}$ .

YEH TOOL KYUN. Hamare paas ek pull hai (Step 1) aur hum ek circle dekh rahe hain (Step 1 picture). Centripetal force bridge hai: yeh batata hai ki ek circle ko exist karne ke liye exactly kitni inward force chahiye. Agar pull exactly woh demand supply kare, toh circle stable hai.

PICTURE. Inward pull arrow aur "required turning" arrow ek hi arrow hain. Yahi coincidence circle ko possible banata hai.

Step 3 — Quantum rule: sirf kuch circles allowed hain

KYA. Ab tak koi bhi radius kaam kar sakta tha — classical physics mein har circle allowed hai. Lekin real atoms aisa nahi karte; woh sirf special sizes use karte hain. Bohr ka rule (Bohr model of the atom se) survivors ko pick karta hai: electron ka angular momentum $m v r$ ek tiny fixed amount $ℏ$ ka whole-number multiple hona chahiye.

YEH TOOL KYUN. Bina kisi rule ke jo zyaadatar orbits ko forbid kare, electron spiral karke andar aa jaata aur atom collapse ho jaata. Hume kuch chahiye jo kahe "yahan step karo, aur yahan, lekin beech mein kahin nahi." Angular momentum ka quantization (ek quantum number $n$ ) woh filter hai.

PICTURE. Guitar string socho: sirf whole numbers of waves loop ke around fit hoti hain. $n = 1$ ek loop hai, $n = 2$ do hain, aur beech mein kuch resonate nahi karta.

Step 4 — Radius $r_{n}$ pin down karo

KYA. Ab hamare paas same orbit ke baare mein do facts hain: force balance $(⋆)$ aur quantum speed $v = \frac{n ℏ}{m r}$ . Doosre ko pehle mein dalo aur unknown $r$ har $n$ ke liye ek specific value par lock ho jaata hai.

YEH STEP KYUN. Do equations, do unknowns ( $v$ aur $r$ ). Substituting $v$ ko khatam karta hai aur $r$ akela reh jaata hai — algebra ka yeh kehne ka tarika hai "yeh do rules milakar har shelf par sirf ek radius allow karte hain."

PICTURE. Allowed circles evenly spaced nahi hain — woh $n^{2}$ ke hisaab se bahar ki taraf balloon karte hain.

Step 5 — Energy add karo (kinetic + potential)

KYA. Electron ki total energy $E$ ke do parts hain: kinetic (motion ki energy, hamesha positive) aur potential (attraction ki stored energy, negative kyunki charges saath aana chahte hain).

YEH STEP KYUN. "Energy level" ka matlab hi total energy hai. Ise paane ke liye hume dono pieces combine karne honge — aur parent note mein [!mistake] (yeh sochna ki $E$ positive hai) exactly potential piece bhoolne se aata hai.

PICTURE. Ek bar chart: KE upar point karta hai, PE do baar neeche, aur unka sum zero se neeche land karta hai.

Step 6 — $r_{n}$ daalo → famous $- 13.6/ n^{2}$

KYA. Hamare paas energy $r$ ke terms mein hai (Step 5) aur $r$ $n$ ke terms mein hai (Step 4). Ek ko doosre mein plug karo aur $n$ ke alawa har symbol ek single number mein collapse ho jaata hai.

YEH STEP KYUN. Yahi payoff hai: $r = a_{0} n^{2}$ substitute karna "ek orbit ki energy" ko "shelf $n$ ki energy" mein badal deta hai — sirf $n$ mein ek formula.

PICTURE. Levels ki ladder: neeche deep, akela $n = 1$ ; rungs $E = 0$ ke just neeche bunch karte hain jaise $n \to \infty$ .

Step 7 — Edge cases (koi scenario unshown mat chhoddo)

KYA AUR KYUN. Ek accha derivation apni extremes pe survive karna chahiye. $n$ ko uski limits tak push karte hain.

Figure poori run dikhata hai: $n = 1$ par deep floor, rungs upar crowd karte hue, $E = 0$ ceiling jise woh approach karte hain lekin kabhi cross nahi karte, aur forbidden $n = 0$ region mark off kiya hua.

Ek-picture summary

Yahan poora chain ek single flow mein compress kiya gaya hai — tum is ek image se poora derivation rebuild kar sakte ho.

Recall Feynman retelling — poora walkthrough simple words mein

Ek proton aur ek electron space mein akele hain. Proton electron ko apni taraf kheenchta hai (Step 1, Coulomb) — aur woh tug exactly woh sideways push hai jo electron ko circle mein bend karne ke liye chahiye (Step 2). Agar nature yahan ruk jaati, toh electron kisi bhi size par circle kar sakta tha aur spiral karke crash ho jaata. Isliye nature ek rule add karti hai (Step 3, Bohr): sirf woh circles jo quantum "loops" ka whole number fit karte hain woh survive karte hain, $n = 1, 2, 3 \dots$ se label kiye hue. Tug-rule ko loop-rule ke saath combine karne se har allowed circle ki size fix ho jaati hai — woh $n^{2}$ ki tarah badhti hain (Step 4). Ab electron ki energy add karo: uski motion energy positive hai, lekin fall-together energy do baar itni negative hai, isliye total hamesha negative nikalta hai — electron ek well mein phansa hai (Step 5). Fixed circle sizes swap karne se woh total ek clean formula ban jaata hai, $- 13.6/ n^{2}$ eV, jahan $13.6$ bas nature ke constants ka ek bundle hai (Step 6). Sabse neecha shelf $n = 1$ sabse gehra hai ( $- 13.6$ eV, escape ke liye $13.6$ eV cost); shelves $0$ ki taraf crowd karti hain jaise tum oopar chahte ho, aur koi $n = 0$ simply exist nahi karta (Step 7). Woh staircase hi atom hai.

Connections

Parent: Hydrogen energy levels
Coulomb's law — andar ki taraf pull (Step 1)
Centripetal force — pull circle kyun banata hai (Step 2)
Bohr model of the atom · Quantum numbers — quantization rule (Step 3)
Rydberg constant and formula — $13.6$ eV kahan se aata hai (Step 6)
Ionization energy — $n = 1 \to \infty$ ki cost (Step 7)
Hydrogen spectral series · Photon energy E = hf = hc/λ — levels kya produce karte hain