Question bank — Blackbody radiation — Planck's quantum hypothesis

False. "Black" ka matlab hai ki woh room temperature par sab kuch absorb karta hai; jab garam hota hai to chamakta hai — Sun aur laal-garam loha near-blackbodies hain jo aankhon ko bilkul bhi kale nahi lagte.

False. Poora point hi universality hai: ek diye hue temperature $T$ par spectrum sirf $T$ ka ek fixed function hota hai, iron, carbon, ya cavity hole ke liye bilkul same.

True. Spectral shape sirf $T$ se decide hoti hai; size aur setting total power aur geometry badlate hain, per-frequency curve shape $u(\nu ,T)$ nahi.

False. Yeh low-frequency limit $h\nu \ll k_{B}T$ mein ekdum sahi hai, jahan Planck's formula isi mein reduce ho jaata hai — yeh sirf high $\nu$ par (catastrophically) fail karta hai.

Fixed $\nu$ par True: zyada $T$ se $h\nu /k_{B}T$ chhota ho jaata hai, to $⟨E⟩\to k_{B}T$. Classical world matlab "quantum lumps thermal energy ke comparison mein tiny hain" wali limit hai.

False. Wien's law ${\lambda }_{\text{peak}}T=b$ ka matlab hai zyada $T$ se chhoti peak wavelength milti hai — zyada garam cheezein peak par bluer glow karti hain, redder nahi.

False. Stefan–Boltzmann kehta hai power $\propto T^4$ hai, to $T$ double karne se total power $2^4=16$ guna ho jaati hai.

True. Jab $h\nu \gg k_{B}T$ hota hai, to "$-1$" $e^{h\nu /k_{B}T}$ ke samne negligible hai, to $⟨E⟩\to h\nu e^{-h\nu /k_{B}T}$, exactly Wien's exponential tail.

False. Yeh classical theory ki failure hai, nature ki nahi. Real ovens Planck's law follow karte hain aur finite amount radiate karte hain; infinity signal karta hai ki theory galat thi.

Error yeh hai ki assume kiya gaya ki har mode ko $k_{B}T$ milta hai. High-$\nu$ modes ko start hone ke liye ek poora quantum $h\nu \gg k_{B}T$ chahiye, to woh frozen out ho jaate hain (exponentially suppressed) — isliye koi X-ray flood nahi hoti.

Galat object. Usne energy ko quantize kiya jo har oscillator hold kar sakta hai ($E_n=nh\nu$), allowed frequencies ko nahi. Saari frequencies exist karti rehti hain; sirf unki energies steps mein aati hain.

$⟨E⟩$ factor $e^{-h\nu /k_{B}T}$ ki tarah decay karta hai, jo ${\nu }^2$ growth ko kuchal deta hai. Exponential decay $\nu$ ke kisi bhi power ko hara deta hai, to product peak karta hai aur zero par aa jaata hai.

Yeh same frequency par correspond nahi karti hain. Kyunki $d\nu /d\lambda$ change of variable ke dauran density ko warp karta hai, $\lambda$-peak aur $\nu$-peak alag-alag physical points par aati hain; ek peak ki location ko naively convert karna galti hai.

Equipartition tab hi valid hai jab energy continuous ho aur thermally accessible ho. Quantized oscillators ke liye jahan $h\nu \gtrsim k_{B}T$ hota hai, uska assumption tootta hai, to theorem wahan simply apply nahi hoti.

Anachronism. Planck ne walls mein material oscillators ke energy exchange ko quantize kiya; photon (light khud quantized hai) baad mein aaya, Einstein ke Photoelectric effect kaam se.

Tum saari frequencies par integrate karte ho (total energy curve ke neeche ka area hai), differentiate nahi. Differentiate karna (slope zero set karna) woh hai jis se Wien's peak milti hai, jo ek alag law hai.

Hole mein ghusne wali ray andar kai baar bounce karti hai, nikalne se pehle near-certainty se absorb ho jaati hai, to absorptivity ≈ 1; garam hone par, hole andar trapped equilibrium radiation re-emit karta hai.

Isko hataane se Wien's law milti hai aur low $\nu$ par classical limit miss ho jaati; isko rakhne se $⟨E⟩\to k_{B}T$ jab $h\nu /k_{B}T\to 0$, jo classical regime ko correctly recover karta hai.

Substitution $x=h\nu /k_{B}T$ $T$ ke chaar factors nikaalti hai — teen integrand mein ${\nu }^3$ se aur ek $d\nu$ se — ek pure dimensionless number (${\pi }^4/15$) times $T^4$ bacha deta hai.

Standing waves ki geometry (cavity mein kitni fit hoti hain) ek real, sahi classical result hai; sirf energy per mode galat thi, to Planck ne count rakh liya aur $k_{B}T$ ko $⟨E⟩$ se replace kar diya.

Thermal equilibrium mein kisi state ki probability uski energy ke saath exponentially girती hai; zyada upar ke rungs $E_n=nh\nu$ increasingly unlikely hote hain, jo exactly costly high-$\nu$ quanta ko freeze out karta hai.

Ek quantum $h\nu$ cost karta hai; jab woh $k_{B}T$ ke comparison mein chhota hota hai to steps continuous lagte hain (classical), lekin jab $h\nu \gg k_{B}T$ hota hai to ek bhi quantum unaffordable hai, to mode dark rehta hai.

Kyunki iska spectrum sirf $T$ par depend karta hai aur kisi par nahi, yeh ek universal function hai — light aur heat ki koi bhi theory isko exactly reproduce karni chahiye, errors chhupane ki koi jagah nahi hoti.

$h\nu /k_{B}T\to \infty$, to $e^{h\nu /k_{B}T}\to \infty$ aur $⟨E⟩\to 0$ — thanda cavity koi thermal quanta hold nahi karta, har mode apni ground state mein hoti hai.

Quantum $h\nu \to 0$, to $e^{h\nu /k_{B}T}-1\to h\nu /k_{B}T$ aur $⟨E⟩\to k_{B}T$ — classical equipartition value, jo dikhata hai ki zero-frequency modes fully classical hoti hain.

Nahi. Yeh zero sirf limits $\nu \to 0$ aur $\nu \to \infty$ mein approach karta hai; har finite positive $\nu$ ke liye value strictly positive hai, to curve beech mein axis ko kabhi touch nahi karti.

Nahi. Frequency peak apna khud ka relation ${\nu }_{\text{peak}}/T=\text{const}$ follow karta hai ek alag numerical constant ke saath, kyunki variables ka change curve ko reshape karta hai — dono peaks genuinely alag physical points hain.

Infinite — ${\int }_0^{\infty }{\nu }^{2}d\nu$ diverge karta hai. Yahi divergence hai ultraviolet catastrophe aur sabse clear sign hai ki classical energy assumption ko abandon karna padega.

Classical Rayleigh–Jeans law apni catastrophe ke saath: $h\to 0$ quanta ko infinitesimal bana deta hai to $⟨E⟩\to k_{B}T$ har jagah, (galat) classical result restore karta hai — quantization exactly $h\ne 0$ effect hai.

Bilkul crossover par — $⟨E⟩=k_{B}T/(e-1)\approx 0.58k_{B}T$, pehle se classical $k_{B}T$ se noticeably neeche, to quantization bite karna shuru kar raha hai par mode ko abhi tak freeze out nahi kiya hai.