Spectral series — Lyman, Balmer, Paschen: Visual walkthrough

2.3.15 · D2 · Physics › Modern Physics › Spectral series — Lyman, Balmer, Paschen

Hum ek sawaal ka jawaab dhundhne wale hain:

Hydrogen sirf kuch fixed colours mein hi kyun chamakta hai, aur woh rang fix kaise hote hain?

Yeh sawaal dimag mein rakho. Neeche ka har step uss jawaab ki taraf ek chhota move hai.

Step 1 — Energy staircase draw karo

KYA. Hydrogen atom ke andar electron kahin bhi nahi baith sakta. Usse sirf kuch specific "floors" par rehne ki permission hai, jinhe ek whole number $n = 1, 2, 3, \dots$ se label kiya jaata hai. Hum $n$ ko principal quantum number kehte hain — ise simply floor number samjho.

KYUN. Yeh "sirf-kuch-floors" rule Quantization of Angular Momentum se aata hai jo Bohr Model of the Atom ke andar hai: electron ki angular momentum sirf ek tiny unit ki whole-number multiples ho sakti hai, aur yahi floors ko discrete hone par majboor karta hai. Nature floor $n = 2.5$ offer nahi karta.

PICTURE. Figure dekho. Floors evenly spaced nahi hain — upar jaane par woh ek doosre ke paas aate jaate hain. Bottom floor $n = 1$ baaki sab se kaafi neeche hai; upar ke floors ek ceiling ki taraf squeeze hote jaate hain.

Figure — Spectral series — Lyman, Balmer, Paschen

Step 2 — Har floor ko ek number do (uski energy)

KYA. Hum har floor ke saath ek energy value attach karte hain: $E_{n} = - \frac{13.6 eV}{n ^{2}}$

Is symbol ko symbol-by-symbol padho, jahan woh baitha hai:

$E_{n}$ — floor number $n$ ki energy. Electron-volts ( $eV$ ) mein measure hoti hai, ek tiny energy unit.
minus sign — kehta hai electron trapped hai (bound). Use free hone ke liye energy add karni padegi.
$13.6 eV$ — ek fixed number, Rydberg energy, ground floor ki depth. (Iska origin hydrogen ki Ionization Energy hai.)
$n^{2}$ — floor number squared. Kyunki yeh denominator mein hai, bada $n$ fraction ko size mein chhota banata hai, isliye upar ke floors kam negative hain — top ke zyada paas.

SQUARED KYUN, aur NEGATIVE KYUN? $n^{2}$ hi woh cheez hai jo floors ko top ke paas squeeze karti hai (Step 1 ki picture): $E_{1} = - 13.6$ , $E_{2} = - 3.4$ , $E_{3} = - 1.51$ , $E_{4} = - 0.85$ — gaps shrink hoti hain. Negative sign "ground level ke neeche trapped" wala bookkeeping hai: level $n \to \infty$ par $E = 0$ milti hai, "free" ceiling.

PICTURE. Wahi staircase, ab har floor par ek value likhi hui. Floor ki height neeche se hai hi uski energy value.

Step 3 — Electron girta hai, aur light paida hoti hai

KYA. Upar ke floor $n_{2}$ par baitha electron neeche ke floor $n_{1}$ par gir sakta hai ( $n_{2} > n_{1}$ ke saath). Jab woh girta hai, atom light ka ek particle — ek photon — bahar fenkta hai.

PHOTON KYUN? Energy conserve honi chahiye. Electron girne se energy khoota hai; woh khoi hui energy gum nahi ho sakti, isliye woh light banke nikl jaati hai. Yeh exactly Emission vs Absorption Spectra ki story hai: emission = girna aur light dena.

PICTURE. Upar ke floor $n_{2}$ se neeche ke floor $n_{1}$ tak ek arrow, ek wavy photon side se bahar shoot karte hue. Drop ki length photon ki energy hai.

Step 4 — Drop measure karo (do floor-values subtract karo)

KYA. Release hui energy do floors ka difference hai: $Δ E = E_{n_{2}} - E_{n_{1}} = - 13.6 (\frac{1}{n _{2}^{2}} - \frac{1}{n _{1}^{2}}) eV .$

Ise in place padho:

$Δ E$ — "energy mein change", drop ki size (Step 3 mein arrow ki length).
$E_{n_{2}} - E_{n_{1}}$ — (top par energy) minus (bottom par energy). Literally "kitni height khoi".
common $- 13.6$ bahar nikalne par andar do fractions milte hain.

Ab awkward minus khatam karne ke liye bracket ke andar order flip karo (yeh sirf algebra hai: $- (a - b) = b - a$ ): $Δ E = 13.6 (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}) eV$

FLIP KYUN? Kyunki $n_{1} < n_{2}$ matlab $\frac{1}{n _{1}^{2}} > \frac{1}{n _{2}^{2}}$ , isliye bracket positive hai. Energy bahar aati hai — ek positive amount. Agar yahan kabhi negative number mile, toh galat order mein subtract kiya hai (mistake box dekho).

PICTURE. Do floor-heights do horizontal levels ke roop mein draw ki gayi hain, aur ek bracket unke beech vertical gap measure kar raha hai — woh gap $Δ E$ hai.

Step 5 — Energy ko colour (wavelength) mein badlo

KYA. Photon ki energy aur uski wavelength Photon Energy and Planck's Relation se ek doosre se locked hain: $E_{photon} = \frac{h c}{λ} .$

$λ$ — wavelength, light wave ki ek ripple ki length (vacuum mein, jaise upar agree hua). Short $λ$ = blue/UV, long $λ$ = red/IR.
$h$ — Planck's constant, nature ka ek fixed number.
$c$ — vacuum mein light ki speed, yeh bhi fixed.
$h c$ — ek fixed bundle; sirf $λ$ ek photon se doosre photon tak badalta hai.

Note karo: energy upar hai, $λ$ neeche. Isliye zyada energy ⇒ chhoti wavelength. Yeh ek fact poori colour ordering explain karta hai.

YEH TOOL KYUN, AUR KOI KYUN NAHI? Humein ek energy (jo Step 4 mein compute ki) se colour (jo ek aanh ya spectrometer measure karta hai) tak bridge chahiye. Planck's relation exactly wahi bridge hai — yeh wahi equation hai jo energy ko seedha wavelength mein convert karti hai.

PICTURE. Do wavy lines: ek tightly-packed high-energy wave (short $λ$ ) aur ek stretched-out low-energy wave (long $λ$ ), arrow ke saath " $E$ up ⇒ $λ$ down".

Step 6 — Do hisson ko jodo

KYA. Release hui energy (Step 4) hai hi photon ki energy (Step 5). Unhe equal set karo: $\frac{h c}{λ} = 13.6 eV (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}) .$

Ab $1/ λ$ akela karne ke liye dono sides ko $h c$ se divide karo:

R_H \;=\; \frac{13.6\ \text{eV}}{hc}.$$ **Units ka dhyan rakho — pehle eV ko joules mein badlo.** Bundle $R_H = 13.6\ \text{eV}/(hc)$ sirf tab "per metre" mein aata hai jab top aur bottom **same** energy unit use karein. Planck's constant aur $c$ SI mein hain ($h$ $\text{J·s}$ mein, $c$ $\text{m/s}$ mein, isliye $hc$ $\text{J·m}$ mein hai), jabki $13.6$ eV mein hai. Isliye pehle convert karo $1\ \text{eV} = 1.602\times10^{-19}\ \text{J}$ se: $$R_H \;=\; \frac{(13.6)(1.602\times10^{-19}\ \text{J})}{(6.626\times10^{-34}\ \text{J·s})(2.998\times10^{8}\ \text{m/s})} \;\approx\; 1.097\times10^{7}\ \text{m}^{-1}.$$ Units cancel hote dekho: $\dfrac{\text{J}}{\text{J·s}\cdot\text{m/s}}=\dfrac{\text{J}}{\text{J·m}}=\dfrac{1}{\text{m}}$. Clean "per metre", exactly ek wavenumber. (Yeh $1.097\times10^7$ mein Step 2 ke footnote se reduced-mass correction already hai.) Master equation ko in place padho: - $\dfrac{1}{\lambda}$ — ==wavenumber== (ripples per metre, vacuum mein). Hum $\lambda$ ki jagah $1/\lambda$ rakhte hain kyunki energy $1/\lambda$ mein **linear** hai, jo formula ko clean rakhta hai. - $R_H$ — **Rydberg constant**, saare fixed numbers ka pile ($13.6\,\text{eV}$ joules mein convert, $h$, $c$) ek mein squash karke. - $\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$ — woh akela part jo badalta hai: yeh purely is par depend karta hai ki *kaunse do floors* ke beech electron jump karta hai. **YEH HUMARA SAWAAL KA JAWAAB KYUN HAI.** Kyunki $n_1,n_2$ whole numbers hain, bracket sirf **kuch certain discrete values** le sakta hai. Isliye $1/\lambda$ — aur therefore colour — sirf kuch certain discrete values ho sakti hai. **Yahi wajah hai ki hydrogen fixed colours mein chamakta hai.** Heavier one-electron ion ke liye, $Z^2$ se multiply karo — dekho [[Hydrogen-like Ions and Z dependence]]. **PICTURE.** Left par do floors ek arrow mein feed hote hain jo master formula mein jaata hai, aur formula ek wavelength axis par ek single sharp spectral line bahar fenkta hai. ![[deepdives/dd-physics-2.3.15-d2-s06.png]] --- ## Step 7 — Jumps group karo: named series **KYA.** **Landing floor** $n_1$ fix karo aur uske upar ke saare floors ko uspe girane do. $n_1$ ki har choice ek **family** of lines deti hai — ek spectral series. - $n_1 = 1$: ==Lyman== (ground floor par landa — sabse gehre drops — **ultraviolet**). - $n_1 = 2$: ==Balmer== (floor 2 par landa — medium drops — **visible**, jo hum dekhte hain). - $n_1 = 3$: ==Paschen== (floor 3 par landa — gentle drops — **infrared**). **IS TARAH GROUP KYUN?** Ek series ki har line mein same $\dfrac{1}{n_1^2}$ term hoti hai (same landing strip); sirf $n_2$ badalta hai. Woh common landing energy hi unhe ek family jaisa dikhata hai. **PICTURE.** Staircase teen coloured "landing strips" ke saath aur arrows har ek par girte hue, har strip apne series name aur light region ke saath labeled. ![[deepdives/dd-physics-2.3.15-d2-s07.png]] --- ## Step 8 — Series ke do edge cases **KYA.** Ek series ke andar, do special jumps saari cheez bound karte hain: 1. **First line** — *sabse chhota* jump, $n_2 = n_1 + 1$. Sabse kam energy ⇒ **sabse lamba** $\lambda$. 2. **Series limit** — *sabse bada* jump, $n_2 \to \infty$. Tab $\dfrac{1}{n_2^2}\to 0$, isliye $$\frac{1}{\lambda} \;=\; \frac{R_H}{n_1^{2}} \quad\Rightarrow\quad \text{series ka sabse chhota } \lambda.$$ **$n_2\to\infty$ SABSE *CHHOTI* WAVELENGTH KYUN DETA HAI (woh sneaky wala).** Jaise $n_2$ badhta hai, term $1/n_2^2$ shrink hoke $0$ ki taraf jaata hai, isliye bracket apni maximum value $1/n_1^2$ tak *grow* karta hai. Bada bracket ⇒ bada $1/\lambda$ ⇒ **sabse chhota** $\lambda$. Series limit ke baad electron "free" ceiling se aata hai — woh essentially unbound tha (yeh ceiling [[Ionization Energy]] edge hai). **PICTURE.** Ek landing strip do arrows ke saath: ek tiny hop abhi upar se (sabse lamba $\lambda$) aur ek giant plunge ceiling $\infty$ se (sabse chhota $\lambda$, the limit). ![[deepdives/dd-physics-2.3.15-d2-s08.png]] > [!mistake] "$n_2=\infty$ sabse lambi wavelength deta hai" > **Kyun sahi lagta hai:** infinity "sabse bada λ" jaisa feel hota hai. > **Fix:** infinity = **sabse bada energy gap** = **sabse chhota** λ. Sabse lamba λ woh *tiny* hop > $n_2=n_1+1$ hai. --- ## Ek-picture summary Poora derivation ek single frame mein: staircase → drop → photon → glued formula → ek real spectrum. ![[deepdives/dd-physics-2.3.15-d2-s09.png]] > [!formula] Sab kuch ek line mein > $$\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right),\quad n_2>n_1,\quad R_H\approx1.097\times10^7\ \text{m}^{-1}\ (\lambda \text{ in vacuum}).$$ > [!recall]- Feynman: poori kahani simple words mein > Atom ke andar ek staircase hai, aur steps top ke paas squish ho jaate hain. Ek tiny ball (electron) > sirf steps par khad ho sakti hai, kabhi beech mein nahi — yahi quantum rule hai. Har step ka ek > "depth" number hota hai, bottom par sabse zyada negative. Jab ball **neeche** koodti hai, woh depth > jo woh khooti hai woh light ki ek flash banke ud jaati hai. Bada drop high-energy flash banata hai > (short, bluish/UV wavelength); chhota drop gentle flash banata hai (long, reddish/IR wavelength). > Kyunki steps *fixed* heights par hain, drops *fixed* sizes ke hote hain, isliye flashes *fixed* colours > ki hoti hain — yahi wajah hai ki hydrogen hamesha same colours mein chamakta hai (vacuum mein > measured). Agar hum saari flashes ko **kaunse step par land karti hain** us hisaab se sort karein, > toh families milti hain: step 1 par land karo → Lyman (UV), step 2 → Balmer (visible wale), step 3 > → Paschen (IR). Har family mein, sabse chhota hop sabse lambi wavelength banata hai, aur bilkul top > (ceiling) se ek plunge sabse chhoti banata hai — series limit, ball ke free hone ki edge par. Ek > chhoti si sachchi baat: proton sach mein nailed down nahi hai, isliye exact staircase numbers ek > "reduced mass" use karti hain — ek $0.05\%$ nudge jo measured $13.6\ \text{eV}$ aur $R_H$ mein > already hai. --- ## Active recall Floors ko kaun sa quantum number label karta hai ::: $n$, the principal quantum number ($n=1,2,3,\dots$) $E_n$ negative kyun hai ::: electron bound (trapped) hai; use free karne ke liye energy add karni padegi Bracket mein landing level $n_1$ pehle kyun rakhte hain ::: taaki $\frac{1}{n_1^2}-\frac{1}{n_2^2}$ positive ho (emission energy release karta hai) Energy ko wavelength mein convert karne wala kaun sa relation hai ::: $E=hc/\lambda$ (Planck) $\lambda$ ki jagah $1/\lambda$ kyun rakhte hain ::: energy $1/\lambda$ mein linear hai, formula clean rehta hai $R_H$ compute karne ke liye kaun sa unit conversion chahiye ::: $13.6\ \text{eV}$ ko joules mein convert karo ($1\ \text{eV}=1.602\times10^{-19}\ \text{J}$) taaki $13.6\,\text{eV}/(hc)$ mein J cancel ho $R_H$ physically kya hai ::: bundle $13.6\,\text{eV}/(hc)\approx1.097\times10^7\ \text{m}^{-1}$ Wavelengths air mein hain ya vacuum mein ::: vacuum wavelengths $13.6$ eV ki value infinite-mass Bohr value se thodi kam kyun hai ::: reduced-mass correction (finite proton mass), lagbhag $0.05\%$ Ek series mein sabse lambi wavelength kaun sa jump deta hai ::: sabse chhota hop, $n_2=n_1+1$ Sabse chhoti wavelength (series limit) kaun sa jump deta hai ::: $n_2\to\infty$, jo $1/\lambda=R_H/n_1^2$ deta hai --- ## Connections - [[Bohr Model of the Atom]] — quantized floors kahan se aate hain. - [[Quantization of Angular Momentum]] — woh rule jo discrete floors force karta hai. - [[Photon Energy and Planck's Relation]] — energy↔wavelength bridge (Step 5). - [[Emission vs Absorption Spectra]] — girna = emission (Step 3). - [[Ionization Energy]] — $E=0$ ceiling aur series limit. - [[Hydrogen-like Ions and Z dependence]] — $R_H\to R_H Z^2$ generalize karta hai. - [[2.3.15 Spectral series — Lyman, Balmer, Paschen (Hinglish)|Yeh note Hinglish mein padho →]]