Hydrogen energy levels Eₙ = −13.6 - n² eV: Foundations

2.3.14 · D1 · Physics › Modern Physics › Hydrogen energy levels Eₙ = −13.6 - n² eV

Yeh page yeh assume karta hai ki aap kuch nahi jaante. Hum har woh symbol, term, aur idea ikathe karte hain jis par parent note Hydrogen energy levels depend karta hai, aur har ek ko ek picture se build karte hain. Upar se neeche padho — har block sirf wahi cheezein use karta hai jo uske upar define ki gayi hain. Hum famous energy formula tab tak nahi likhenge jab tak humne usmein har letter earn nahi kar liya (woh §7 mein hoga).

0 · Characters ka cast (atom khud)

Figure 1 — Hydrogen atom. Electron proton ke around chakkar lagata hai; dono ke beech ki pull hi poori kahani hai.

Figure — Hydrogen energy levels Eₙ = −13.6 - n² eV

Figure 1 dekhein: coral dot proton hai, lavender dot electron hai jo circular path par hai. Is topic mein sab kuch is baare mein hai ki woh electron kitna tightly held hai aur kaunse circles par travel karne ki use permission hai.

1 · Charge $e$ , distance $r$ , aur electric pull $F$

2 · Speed $v$

3 · Mass $m$ aur "turning" force

Recall A small honesty note: reduced mass

Hum proton ko bilkul fixed treat karte hain aur electron mass $m$ use karte hain. Sach mein proton bhaari hai par infinitely bhaari nahi, isliye dono particles apne shared centre ke baare mein thoda hiltey hain. Iska exact fix yeh hai ki $m$ ki jagah reduced mass $μ = \frac{m M}{m + M}$ use karein, jahan $M$ proton mass hai. Kyunki $M \approx 1836 m$ , $μ$ sirf $m$ se lagbhag $0.05%$ chhota hai — itna ki $13.6$ eV ko fourth digit mein shift kar de, par jo kuch hum yahaan karte hain usmein dikh nahi sakta. Apne saare kaam ke liye $m$ rakhein; bas yeh jaano ki $μ$ precise version hai.

Figure 2 — Do forces, ek direction. Electric pull aur centripetal need dono inward point karte hain, isliye hum unhe equal set karte hain.

4 · Kinetic aur potential energy

Figure 3 — Energy well. Zero energy rim hai (electron escape hua); koi bhi bound electron uske neeche baitha hai, isliye uski total energy negative hai.

5 · Pure whole numbers $n$ (shelves)

6 · Do rules ko ek saath daalna — Bohr radius $a_{0}$

Ab hamare paas do ready-made facts hain. Rule A (§3 se force balance): Coulomb pull centripetal need ke barabar hai, $\frac{1}{4 π ε _{0}} \frac{e ^{2}}{r ^{2}} = \frac{m v ^{2}}{r} .$ Rule B (§5 se quantization): $m v r = n ℏ$ .

Formula

$r_{n} = a_{0} n^{2}$ step by step derive karna Pehle kya karte hain: Rule A ko tidy karo dono sides ko $r$ se multiply karke, taaki messy $r^{2}$ ek single $r$ ban jaaye: $m v^{2} = \frac{1}{4 π ε _{0}} \frac{e ^{2}}{r} . (⋆)$ Kyun: yeh $m v^{2}$ ko isolate karta hai, jise hum Rule B use karke attack kar sakte hain. Aage, Rule B se speed $v$ solve karo (taaki hum use eliminate kar sakein): $v = \frac{n ℏ}{m r} .$ Kyun: hamare paas do unknowns hain ( $v$ aur $r$ ); $v$ hatane se $r$ akele mein ek equation milti hai. Ab substitute karo yeh $v$ ko $(⋆)$ mein:

\;\Longrightarrow\; \frac{n^2\hbar^2}{m r^2} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r}.$$ **Aakhir mein**, dono sides ko $r^2$ se multiply karo aur right-hand coefficient se divide karo $r$ solve karne ke liye: $$\boxed{\,r_n = \frac{4\pi\varepsilon_0\,\hbar^2}{m e^2}\;n^2 = a_0\,n^2\,},\qquad a_0 = \frac{4\pi\varepsilon_0\,\hbar^2}{m e^2}=0.529\ \text{Å}.$$ **Iska matlab:** $n^2$ ke aage constants ka woh poora clump *hi* Bohr radius $a_0$ hai; orbits $n^2$ ke saath badte hain, isliye electron upar wali shelves par baahir baithta jaata hai.

Centre par ek fixed positive charge

+ e

jo electron ko andar kheenchta hai.

e

kya hai?

Elementary charge,

1.6 \times 1 0^{- 19}

C, proton (

+

) aur electron (

-

) dwara carry kiya jaata hai.

Factor

\frac{1}{4 π ε _{0}}

kis kaam ka hai?

Woh constant jo Coulomb's law mein charges aur distance ko force mein convert karta hai.

r

yahaan kya matlab hai?

Electron ki circular orbit ki radius (proton-to-electron distance).

F

kya hai?

Newtons mein force — yahaan proton ka electron par electric pull, Coulomb's law se diya gaya.

Centripetal force kya hai aur uska formula kya hai?

Kisi mass ko circle par rakhne ke liye zaroori inward pull,

m v^{2} / r

Hum Coulomb pull ko centripetal force ke barabar kyun set karte hain?

Electric attraction exactly woh pull supply karta hai jo circle ko chahiye.

Kinetic energy kya hai?

Motion ki energy,

\frac{1}{2} m v^{2}

; hamesha positive.

Yahaan potential energy kya hai aur uska sign?

Position ki stored energy,

- \frac{1}{4 π ε _{0}} \frac{e ^{2}}{r}

; negative, zero infinity par chosen hai.

Total energy

E

(aur

E_{n}

) kya hai?

E = KE + PE

; jab orbit number

n

tak restrict ho toh yeh

E_{n}

ban jaata hai.

Total energy negative kyun hoti hai?

Kyunki deep negative PE positive KE se zyada hai — electron ek well mein trapped hai.

Reduced mass kya hai aur iska zikr kyun karte hain?

μ = m M / (m + M)

; exact accuracy ke liye

m

ki jagah use hota hai kyunki proton bhaari hai par infinitely bhaari nahi.

Principal quantum number

n

kya hai?

Ek whole number

1, 2, 3, \dots

jo allowed orbits/energies ko label karta hai.

Quantization rule kya hai?

Angular momentum

m v r = n ℏ

n

ko integer hone par majboor karta hai.

ℏ

kya hai?

Planck's constant divided by

2 π

ℏ = h /2 π

Bohr radius

a_{0}

kya hai?

Sabse chhoti allowed orbit radius,

a_{0} = 0.529

Å, isliye

r_{n} = a_{0} n^{2}

KE aur PE milkar

E_{n}

kaise dete hain?

E = \frac{1}{2} k e^{2} / r - k e^{2} / r = - \frac{1}{2} k e^{2} / r = - KE

;

r = a_{0} n^{2}

substitute karo to

- 13.6/ n^{2}

eV milta hai.

Speed of light

c

kya hai?

Light vacuum mein kitni tezi se travel karti hai,

3.0 \times 1 0^{8}

m/s, har colour ke liye same.

Δ E

kya hai?

Do levels ke beech energy difference,

E_{n_{i}} - E_{n_{f}}

, jo ek photon ban jaata hai.

Ek electron-volt kya hai?

1.6 \times 1 0^{- 19}

J — atom-sized energy unit.

Wavelength ke terms mein photon ki energy kya hai?

E_{photon} = h c / λ

, aur quick conversion ke liye

h c = 1240

eV·nm.

Connections

Bohr model of the atom — quantization postulate $m v r = n ℏ$ supply karta hai.
Coulomb's law, Centripetal force — do forces Step 1 mein equal set ki gayi hain.
Ionization energy, Quantum numbers — well depth aur $n$ ka matlab.
Photon energy E = hf = hc/λ, Hydrogen spectral series, Rydberg constant and formula — jahan levels aage le jaate hain.

Foundations — Hydrogen energy levels Eₙ = −13.6 - n² eV

0 · Characters ka cast (atom khud)

1 · Charge $e$ , distance $r$ , aur electric pull $F$

2 · Speed $v$

3 · Mass $m$ aur "turning" force

4 · Kinetic aur potential energy

5 · Pure whole numbers $n$ (shelves)

6 · Do rules ko ek saath daalna — Bohr radius $a_{0}$

7 · Radius se energy formula $E_{n}$ tak

8 · Light: speed $c$ , photon, aur $λ$

Yeh foundations topic ko kaise feed karte hain

Equipment checklist

Connections