2.3.1
Intuition Ek-sentence ki kahaniEk garam object glow karta hai, aur kaise glow karta hai (colours ka spectrum) classical physics ko follow karne se mana kar diya. Planck ne ise ek wild guess se fix kiya: ==energy of oscillators is quantized in lumps of h ν
Ek blackbody ek idealized object hai jo absorbs all radiation falling on it (kuch reflect nahi karta) aur, jab garam hota hai, ==emits a characteristic spectrum that depends only on its temperature T
YEH kyun important hai: Kyunki emit kiya hua spectrum sirf T depend karta hai, yeh ek universal function hai — fundamental physics ke liye ek perfect testbed. Ek cavity mein chhota sa hole real-world ka sabse accha blackbody hai: andar jaane wali koi bhi ray bounce karti rahti hai aur bahar nikalne se pehle absorb ho jaati hai.
Intuition Hole = blackbody kyunEk photon jo kisi cave mein ghus jaata hai, use waapas bahar aane ka raasta lagbhag kabhi nahi milta — woh walls par absorb ho jaata hai. Isliye hole bilkul black dikhta hai. Jab cavity garm hoti hai, toh hole andar trapped equilibrium radiation emit karta hai.
Spectral curve u ( ν , T )
Wien's displacement law: peak wavelength temperature ke saath shift hoti hai: λ peak T = b b ≈ 2.898 × 1 0 − 3 m⋅K Stefan–Boltzmann law: total emitted power ∝ T 4
Definition Rayleigh–Jeans law (classical)Ek cavity mein electromagnetic standing-wave modes count karke aur har ek ko classical average energy k B T u ( ν , T ) d ν = c 3 8 π ν 2 k B T d ν
YEH kaise milta hai (first principles se):
[ ν , ν + d ν ] c 3 8 π ν 2 d ν Classical equipartition: har mode (ek harmonic oscillator) ki average energy ⟨ E ⟩ = k B T
Multiply karo: energy density = (modes)×(energy per mode).
Common mistake Steel-man: "Har mode kok B T Kyun sahi lagta hai: Equipartition ek rock-solid classical theorem hai — har quadratic degree of freedom ko 2 1 k B T k B T Catastrophe: Modes ν 2 u → ∞ ν → ∞ ∫ 0 ∞ ν 2 d ν = ∞ Fix: Galti mode-counting mein nahi hai — galti yeh assume karne mein hai ki har mode koi bhi (continuous) amount of energy rakh sakti hai. Planck ne energy ko quantize kiya.
Definition Planck's postulateFrequency ν discrete steps mein energy rakh sakta hai:
E n = nh ν , n = 0 , 1 , 2 , … h = 6.626 × 1 0 − 34 J⋅s Planck's constant hai. Energy h ν
Intuition Yeh catastrophe ko kyun khatam karta haiLow ν h ν ≪ k B T ⟨ E ⟩ → k B T High ν h ν ≫ k B T itni energy maangta hai jo thermally available nahi hai, isliye woh modes "frozen out" ho jaate hain — e − h ν / k B T
Worked example Stefan–Boltzmann (∝ T 4 U = ∫ 0 ∞ u ( ν , T ) d ν x = h ν / k B T ν = h k B T x d ν = h k B T d x Kyun substitute karte hain? Saari T U = c 3 8 π h ( h k B T ) 4 ∫ 0 ∞ e x − 1 x 3 d x = π 4 /15 U ∝ T 4 4th power numerator mein ν 3 d ν 1/ h 3 T
Worked example Wien's displacement law — Yeh step kyun?u ( λ , T ) ∂ λ ∂ u = 0 y = h c / λ k B T Kyun? Peak wahan hoti hai jahan slope zero ho. Differentiate karne par transcendental equation milti hai:
5 ( 1 − e − y ) = y ⇒ y ≈ 4.965 λ peak T = 4.965 k B h c = b ≈ 2.898 × 1 0 − 3 m⋅K
Worked example Numerical: Sun ka temperatureSun ka spectrum λ peak ≈ 500 nm T = λ peak b = 500 × 1 0 − 9 2.898 × 1 0 − 3 ≈ 5800 K Yeh kyun kaam karta hai: Wien's law ek measurable peak colour ko surface temperature se link karta hai — astronomers exactly yahi use karte hain.
Intuition Agar sirf yahi yaad rakhna hai
Classical = k B T
Planck: energy h ν ν ⟨ E ⟩ = e h ν / k B T − 1 h ν
Sab kuch (Planck law, Stefan T 4 λ p e ak T = b
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho ek piano hai jahan tum sirf poore keys daba sakte ho, kabhi "aadha key" nahi. Low notes ke liye keys easy aur sasti hain, isliye warm room unhe bahut dabaata hai — yahi visible glow hai. Lekin bahut high notes dabane ke liye ek baar mein bhi bahut zyada energy chahiye, isliye ek garam room bhi unhe afford nahi kar sakta — woh chup rahte hain. Old physics sochti thi tum key ka koi bhi chhota sa fraction daba sakte ho, isliye usne predict kiya ki room infinitely many ultra-high notes chilla raha hai ("ultraviolet catastrophe"). Planck ne kaha: nahi, energy h ν
Mnemonic Average energy yaad karo"H-nu over EXP minus one" → e h ν / k B T − 1 h ν − 1 quantum signature hai; ise hata do aur classical k B T − 1
Blackbody kya hota hai? Ek ideal body jo saari incident radiation absorb karta hai aur sirf apne temperature par depend karta spectrum emit karta hai.
Ek garam cavity mein chhota sa hole ideal blackbody kyun hai? Andar jaane wali radiation bounce karti hai aur absorb ho jaati hai escape se pehle (absorptivity ≈ 1), isliye yeh equilibrium thermal radiation emit karta hai.
Rayleigh–Jeans law kya hai? u ( ν , T ) d ν = c 3 8 π ν 2 k B T d ν k B T
Ultraviolet catastrophe kya hai? Rayleigh–Jeans predict karta hai u → ∞ ν → ∞
Planck ki hypothesis kya thi? Oscillators sirf E n = nh ν h ν
Planck oscillator ki average energy? ⟨ E ⟩ = e h ν / k B T − 1 h ν
Quantization catastrophe ko kyun rokta hai? High-ν h ν ≫ k B T
Planck's radiation law (frequency form) batao. u ( ν , T ) d ν = c 3 8 π ν 2 ⋅ e h ν / k B T − 1 h ν d ν
Planck's law ka low-frequency limit? Rayleigh–Jeans reduce ho jaata hai, c 3 8 π ν 2 k B T
Planck's law ka high-frequency limit? Wien's law reduce ho jaata hai, c 3 8 π h ν 3 e − h ν / k B T
Wien's displacement law? λ peak T = b ≈ 2.898 × 1 0 − 3
Planck se Stefan–Boltzmann law? Total energy density U ∝ T 4 ∫ x 3 / ( e x − 1 ) d x = π 4 /15
Planck's constant ki value? h ≈ 6.626 × 1 0 − 34
Photoelectric effect — E = h ν Wien's displacement law · Stefan–Boltzmann law — limiting cases yahan derive hue.Equipartition theorem — classical assumption jo fail hoti hai.Bose–Einstein statistics — e x − 1 Cosmic Microwave Background — sabse perfect blackbody known (T = 2.725 Quantum harmonic oscillator — jahan E n = nh ν E n = ( n + 2 1 ) h ν
flaw is continuous energy
Blackbody absorbs all
Mode counting 8pi nu^2 / c^3
Classical equipartition kBT
Planck postulate E=n h nu
