Blackbody radiation from statistical mechanics — Planck distribution

2.4.19 · Physics › Thermodynamics & Statistical Mechanics (Advanced)

1. Hum kya count kar rahe hain? (WHAT)

Side $L$ ke ek cubic cavity ke andar, EM field standing-wave modes ka superposition hai. Har mode ko ek wavevector $k$ aur ek polarization se label kiya jaata hai. Humein do cheezein chahiye:

Kitne modes hain har frequency par — woh hai density of states $g (ν)$ .
Kitni energy har mode average mein carry karti hai temperature $T$ par — $⟨ ϵ ⟩$ .

Spectrum in dono ka product hai: $u (ν) d ν = g (ν) ⟨ ϵ (ν)⟩ d ν$

2. Density of states (HOW — derive karo)

Ye modes kyun? Standing waves walls par vanish honi chahiye, isliye allowed wavevector components quantized hain: $k_{x} = \frac{n _{x} π}{L}, k_{y} = \frac{n _{y} π}{L}, k_{z} = \frac{n _{z} π}{L}, n_{i} = 1, 2, 3, \dots$ Ye step kyun? Woh wave jiska node $x = 0$ aur $x = L$ par ho, usmein half-integer wavelengths fit honi chahiye.

$n$ -space mein modes ek cubic lattice banate hain spacing 1 ke saath. $n$ tak magnitude wale modes ki number ek sphere ke ek octant ka volume hai (kyunki $n_{i} > 0$ ): $N (n) = \frac{1}{8} \cdot \frac{4}{3} π n^{3} = \frac{π n ^{3}}{6}$ Ek-aathwan kyun? Sirf positive $n_{i}$ count hote hain — woh full sphere ka $1/8$ hai.

Ab $n$ ko frequency mein convert karo. $∣ k ∣ = nπ / L$ aur $∣ k ∣ = 2 π ν / c$ ke saath: $n = \frac{2 Lν}{c} \Rightarrow N (ν) = \frac{π}{6} (\frac{2 Lν}{c})^{3} = \frac{4 π L ^{3} ν ^{3}}{3 c ^{3}}$

2 se multiply karo light ke do polarizations ke liye, volume $L^{3}$ se divide karo, aur differentiate karo: $g (ν) = \frac{1}{L ^{3}} \frac{d}{d ν} [2 N (ν)] = \frac{8 π ν ^{2}}{c ^{3}}$

3. Har mode ki average energy (HOW — quantum step)

Classical answer (galat): equipartition har oscillator ko $⟨ ϵ ⟩ = k_{B} T$ deta hai $ν$ ki parwah kiye bina. Tab $u (ν) = \frac{8 π ν ^{2}}{c ^{3}} k_{B} T$ — yeh Rayleigh–Jeans law hai, jo $ν \to \infty$ par diverge karti hai.

Planck's quantization (sahi): frequency $ν$ ke mode ki sirf discrete energies hoti hain $ϵ_{n} = nh ν, n = 0, 1, 2, \dots$

Canonical ensemble use karo. Define karo $β = 1/ (k_{B} T)$ aur $x = β h ν$ . Partition function: $Z = \sum_{n = 0}^{\infty} e^{- nh ν β} = \sum_{n = 0}^{\infty} (e^{- x})^{n} = \frac{1}{1 - e ^{- x}}$ Ye step kyun? Geometric series with ratio $e^{- x} < 1$ .

Average energy:

= \frac{h\nu\, e^{-\beta h\nu}}{1-e^{-\beta h\nu}}$$ Upar aur neeche $e^{\beta h\nu}$ se multiply karo: $$\boxed{\langle\epsilon\rangle = \dfrac{h\nu}{e^{h\nu/k_BT}-1}}$$ Yeh ek quantum oscillator ki average energy hai (zero-point minus — equilibrium *photon* picture mein: yeh Bose–Einstein occupation $\bar n = 1/(e^{h\nu/k_BT}-1)$ times $h\nu$ hai). --- ## 4. Planck distribution (assemble karo) $$\boxed{u(\nu,T)\,d\nu = \frac{8\pi\nu^2}{c^3}\cdot\frac{h\nu}{e^{h\nu/k_BT}-1}\,d\nu = \frac{8\pi h\nu^3}{c^3}\frac{1}{e^{h\nu/k_BT}-1}\,d\nu}$$ ![[2.4.19-Blackbody-radiation-from-statistical-mechanics-—-Planck-distribution.png]] > [!intuition] Quantization catastrophe ko kaise khatam karta hai > Jab $h\nu \gg k_BT$ ho, $e^{h\nu/k_BT}$ bahut bada hota hai, isliye $\langle\epsilon\rangle \approx h\nu\,e^{-h\nu/k_BT}\to 0$. > High-frequency modes *exponentially* suppress ho jaate hain — unhe on hone ke liye bhi poora quantum $h\nu$ chahiye, > aur $k_BT$ size ke thermal kicks woh afford nahi kar sakte. $\nu^2$ ki growth exponential decay se haar jaati hai. --- ## 5. Do limits (Forecast-then-Verify) > [!example] Forecast: $u(\nu)$ low aur high $\nu$ par kya karega? > **Low $\nu$** ($h\nu \ll k_BT$): photons saste → classical $k_BT$ per mode expect karo → Rayleigh–Jeans recover hoga. > **High $\nu$** ($h\nu \gg k_BT$): photons maheenge → exponential cutoff expect karo (Wien). > Ab expand karke verify karo. **Low frequency** ($x = h\nu/k_BT \to 0$): $e^x - 1 \approx x$, isliye $$\langle\epsilon\rangle \approx \frac{h\nu}{h\nu/k_BT} = k_BT \quad\Rightarrow\quad u \approx \frac{8\pi\nu^2}{c^3}k_BT$$ ✓ Rayleigh–Jeans recover ho gaya. *Ye step kyun?* Taylor: $e^x = 1+x+\dots$. **High frequency** ($x\to\infty$): $e^x - 1 \approx e^x$, isliye $$u \approx \frac{8\pi h\nu^3}{c^3}e^{-h\nu/k_BT}$$ ✓ Wien's law — exponential tail. --- ## 6. Worked consequences > [!example] Stefan–Boltzmann from Planck (har line par Why this step?) > Total energy density: saare $\nu$ par integrate karo. > $$U = \int_0^\infty \frac{8\pi h\nu^3}{c^3}\frac{d\nu}{e^{h\nu/k_BT}-1}$$ > Substitute karo $x = h\nu/k_BT$, toh $\nu = \frac{k_BT}{h}x$, $d\nu = \frac{k_BT}{h}dx$ *(kyun? integral ko dimensionless banata hai)*: > $$U = \frac{8\pi h}{c^3}\left(\frac{k_BT}{h}\right)^4 \int_0^\infty \frac{x^3\,dx}{e^x-1}$$ > Integral ek standard result hai $\displaystyle\int_0^\infty\frac{x^3}{e^x-1}dx = \frac{\pi^4}{15}$ *(kyun? yeh $\Gamma(4)\zeta(4)=6\cdot\pi^4/90$ ke barabar hai)*. > $$U = \frac{8\pi^5 k_B^4}{15 c^3 h^3}T^4 \propto T^4$$ > Toh total radiated power $\propto T^4$ — **Stefan–Boltzmann**. $T$ double karo toh brightness 16× ho jaati hai. > [!example] Wien displacement (peak frequency) > $u(\nu)$ maximize karo: $\frac{d}{dx}\left(\frac{x^3}{e^x-1}\right)=0$ set karo jahan $x=h\nu/k_BT$. > Yeh transcendental equation $3(1-e^{-x}) = x$ deta hai, numerically $x \approx 2.821$ se solve hota hai. > Isliye $\nu_{\max} = 2.821\,\frac{k_BT}{h} \propto T$: zyada garam bodies zyada high frequency par peak karti hain (isliye ek star bahut garam hone par > blue-white dikhta hai, thanda hone par red). --- ## 7. Common mistakes (Steel-man + fix) > [!mistake] "Equipartition hold honi chahiye — har mode ko $k_BT$ milta hai." > **Kyun sahi lagta hai:** equipartition *continuous* classical energies ke liye rigorously true hai. **Flaw:** > yeh secretly assume karta hai ki energy continuous hai. Discrete levels $nh\nu$ ke saath, woh mode jiska gap $h\nu \gg k_BT$ ho > mostly apne ground state mein stuck rehta hai aur $k_BT$ se kaafi kam energy carry karta hai. **Fix:** $\langle\epsilon\rangle$ > ko *quantized* partition function se compute karo — woh $k_BT$ sirf $h\nu\ll k_BT$ limit mein deta hai. > [!mistake] Polarization ke liye factor of 2 bhool jaana. > **Kyun sahi lagta hai:** tumne saare wavevectors carefully count kiye, complete lagta hai. **Flaw:** har $\vec k$ > do independent transverse polarizations support karta hai. **Fix:** $g(\nu)$ ko 2 se multiply karo → $8\pi\nu^2/c^3$ (4 galat hoga, missing 2 Stefan constant ko half kar dega). > [!mistake] $u(\nu)$ peak aur $u(\lambda)$ peak ko confuse karna. > **Kyun sahi lagta hai:** "peak toh peak hai." **Flaw:** $u(\nu)d\nu = u(\lambda)d\lambda$ aur > Jacobian $|d\nu/d\lambda| = c/\lambda^2$ maximum shift kar deta hai. **Fix:** $x_\nu\approx2.82$ lekin $x_\lambda\approx4.97$; > woh *same* spectrum ko alag variables mein describe karte hain. --- > [!recall]- Feynman: 12-saal ke bachche ko explain karo > Socho ek room mein alag-alag lengths ki jump-ropes bhari hain. Slow, lambi ropes ko swing karana easy hai — thodi > si energy unhe shuru kar deti hai. Super-fast choti ropes ko ek saath ek bada dhakka chahiye hila ne ke liye; chote taps se kuch nahi hota. > Heat random chote taps deti hai. Toh slow ropes khushi se swing karti hain, lekin fast wale mostly still rehte hain. > Isliye ek warm object mostly certain colors par glow karta hai aur kabhi infinitely-blue light par nahi — fast > "ropes" shuru karna bahut mushkil hai, isliye woh chup rehti hain. Room ko zyada garam karo (bade taps) aur fast ropes > bhi join kar leti hain, toh glow blue ki taraf shift hota hai. > [!mnemonic] **"Count modes, quantize energy, the catastrophe dies."** > $g\sim\nu^2$ (count) × $\langle\epsilon\rangle\sim h\nu/(e^{x}-1)$ (quantize) ⇒ exponential tail $\nu^2$ ko beat karta hai. > Combo ko yaad rakho **"$\nu^3$ up, $e^{x}$ down"** ke taur par. --- ## Flashcards #flashcards/physics Statistically, ek blackbody ko kis physical system se model kiya jaata hai? ::: EM standing-wave modes ki cavity (har ek quantum oscillator) jo walls ke saath thermal equilibrium mein ho. EM modes ke liye density of states likho. ::: $g(\nu)=8\pi\nu^2/c^3$ (polarization ke liye ×2 include hai). Mode counting mein $1/8$ ka factor kyun? ::: Allowed $n_x,n_y,n_z>0$, isliye $\vec n$-space mein sphere ka sirf ek octant count hota hai. Planck ne kaunsa quantization impose kiya? ::: Frequency $\nu$ ke mode ki energies $\epsilon_n=nh\nu$, $n=0,1,2,\dots$ hoti hain. Average energy per mode (Planck)? ::: $\langle\epsilon\rangle=\dfrac{h\nu}{e^{h\nu/k_BT}-1}$. Full Planck distribution $u(\nu,T)$ state karo. ::: $u=\dfrac{8\pi h\nu^3}{c^3}\dfrac{1}{e^{h\nu/k_BT}-1}$. $\langle\epsilon\rangle$ ka low-frequency limit? ::: $\to k_BT$ (Rayleigh–Jeans), $e^x-1\approx x$ se. $u(\nu)$ ka high-frequency limit? ::: $\propto \nu^3 e^{-h\nu/k_BT}$ (Wien's law). Classically ultraviolet catastrophe ka kya cause hai? ::: Equipartition har mode ko $k_BT$ deta hai; $g\sim\nu^2$ ke saath integral diverge karta hai. Quantization catastrophe kaise hatata hai? ::: $h\nu\gg k_BT$ modes ko ek poora quantum $h\nu$ chahiye, isliye woh exponentially freeze out ho jaate hain. Stefan–Boltzmann temperature dependence? ::: $U\propto T^4$ ($\int x^3/(e^x-1)dx=\pi^4/15$ se). Wien displacement: peak frequency vs T? ::: $\nu_{\max}\propto T$, jahan $x=h\nu/k_BT\approx2.82$. --- ## Connections - [[Bose-Einstein statistics]] — $\langle n\rangle = 1/(e^{\beta h\nu}-1)$ photon occupation number hai. - [[Partition function]] — $Z=\sum e^{-\beta\epsilon_n}$ poori derivation drive karta hai. - [[Equipartition theorem]] — classical limit aur *kyun* yeh yahan fail hoti hai. - [[Rayleigh-Jeans law]] aur [[Wien's law]] — woh do limits jo Planck ne unify kiye. - [[Stefan-Boltzmann law]] — Planck spectrum ka integral. - [[Density of states]] — same mode-counting phonons (Debye model) aur gases ke liye reuse hoti hai. - [[Photoelectric effect]] — $E=h\nu$ quanta ka independent evidence. ## 🖼️ Concept Map ```mermaid flowchart TD BB[Blackbody cavity] -->|filled with| MODES[EM standing wave modes] MODES -->|each is| OSC[Harmonic oscillator] MODES -->|count gives| GDOS[Density of states g of nu] GDOS -->|grows as nu squared| SEED[Many high-freq modes] OSC -->|classical energy kBT| RJ[Rayleigh-Jeans law] SEED -->|combined with kBT| RJ RJ -->|diverges at high nu| UV[Ultraviolet catastrophe] OSC -->|energy in lumps h nu| PLANCK[Planck quantization] PLANCK -->|freezes high modes| AVGE[Average energy per mode] GDOS -->|times avg energy| SPEC[Spectrum u of nu] AVGE -->|times g of nu| SPEC PLANCK -->|resolves| UV ```