Black-body radiation and Planck's quantum hypothesis E = hν

2.1.1 · Chemistry › Quantum Atomic Structure

WHAT is a black body?

WHY do we care? Kyunki iska emission spectrum universal hai — har garam body (ek star, ek filament, ek furnace) usi curve ko follow karta hai. Agar hum is curve ko explain kar sakein, toh hum thermal light ko khud samajh lete hain.

WHAT the experiment shows

Jab tum fixed temperature par intensity (emitted energy) vs wavelength plot karte ho toh ek humped curve milta hai:

Curve upar jaata hai, kisi $λ_{ma x}$ par peak karta hai, phir neeche aata hai.
Jaise-jaise temperature badhti hai, peak choti wavelength ki taraf shift hoti hai (redhot → whitehot → bluehot). Yahi Wien's displacement law hai: $λ_{ma x} T = constant$ .
Curve ke neeche ka total area (total energy) $T^{4}$ ke saath badhta hai (Stefan–Boltzmann).

Figure — Black-body radiation and Planck's quantum hypothesis E = hν

WHY classical physics FAILED — the Ultraviolet Catastrophe

Classical physics (Rayleigh–Jeans) cavity ko standing electromagnetic waves se bhara maanta tha, har ek wave ek oscillator ki tarah energy equally share karta tha (equipartition: $k T$ per mode).

Disaster yeh tha: jaise $ν \to \infty$ (short wavelength / UV), $u (ν) \to \infty$ . Theory predict karti hai ki high frequency par infinite energy radiate hogi — yahi ultraviolet catastrophe hai. Reality mein curve wahaan zero tak gir jaata hai.

HOW Planck fixed it — the quantum hypothesis

Planck ka radical assumption: frequency $ν$ ka ek oscillator koi bhi energy nahi rakh sakta. Uske paas energy sirf ek basic packet ke whole-number multiples mein ho sakti hai:

Average energy per mode ki derivation (first principles se):

$⟨ E ⟩ = k T$ ki jagah, discrete levels $E_{n} = nh ν$ par Boltzmann weights use karke average compute karo:

$⟨ E ⟩ = \frac{\sum _{n = 0}^{\infty} nh ν e ^{- nh ν / k T}}{\sum _{n = 0}^{\infty} e ^{- nh ν / k T}}$

$x = e^{- h ν / k T}$ maano. Denominator ek geometric series hai: $\sum x^{n} = \frac{1}{1 - x}$ Numerator: $h ν \sum n x^{n} = h ν \frac{x}{( 1 - x ) ^{2}}$ .

Divide karo: $⟨ E ⟩ = h ν \frac{x}{1 - x} = \frac{h ν}{\frac{1}{x} - 1} = \frac{h ν}{e ^{h ν / k T} - 1}$

WHY yeh catastrophe theek karta hai: high $ν$ par, factor $e^{h ν / k T}$ bahut bada ho jaata hai, isliye $⟨ E ⟩ \to 0$ . High-frequency modes frozen out ho jaate hain — unhe excite karne ke liye ek poora packet $h ν$ chahiye, aur $k T$ itna kaafi nahi hota. Curve neeche gir jaata hai, jo experiment se match karta hai.

Classical limit check karo: choti $ν$ (ya badi $T$ ) ke liye, $e^{h ν / k T} \approx 1 + \frac{h ν}{k T}$ , toh $⟨ E ⟩ \approx \frac{h ν}{h ν / k T} = k T$ Rayleigh–Jeans wapas mil jaata hai. Planck's law mein classical result ek special case ke roop mein shamil hai.

Worked Examples

Common Mistakes (Steel-man + fix)

Recall Feynman: 12-saal ke bachche ko explain karo

Ek ramp ki jagah ek staircase imagine karo. Ramp par tum kisi bhi height par khade ho sakte ho; staircase par tum sirf poore steps par khade ho sakte ho. Planck ne kaha energy ek staircase ki tarah hai — ek chota garam atom sirf $h ν$ size ke poore steps mein "jump" kar sakta hai. High notes (high frequency) ke bahut unche steps hote hain, toh ek bhi step chadhna mushkil hai — isliye garam cheezein ultraviolet mein crazily bright nahi glowti. Yeh staircase idea hi hai jis wajah se poori quantum world exist karti hai!

Flashcards

Black body kya hai?

Ek idealized object jo sabhi incident radiation absorb karta hai aur sirf temperature par depend karta hua spectrum emit karta hai.

Ultraviolet catastrophe kya hai?

Classical (Rayleigh–Jeans) prediction ki black-body intensity short wavelengths par → infinity, jo experiment ke ulta hai.

Planck's quantum hypothesis batao.

Oscillator energy quantized hai:

E = nh ν

; sabse chota packet

E = h ν

hai.

Planck's constant ki value aur units?

h = 6.626 \times 1 0^{- 34} J s

Planck's law mein high-frequency modes blow up kyun nahi karte?

Unhe ek poora quantum

h ν ≫ k T

chahiye; Boltzmann factor

e^{- h ν / k T}

unhe starve karta hai, isliye

⟨ E ⟩ \to 0

Planck's model mein average energy per mode?

⟨ E ⟩ = \frac{h ν}{e ^{h ν / k T} - 1}

Planck's law classical result kaise recover karta hai?

h ν ≪ k T

ke liye,

e^{h ν / k T} \approx 1 + h ν / k T

, jisse

⟨ E ⟩ \approx k T

milta hai.

Wavelength se photon energy ka formula?

E = h c / λ

Wien's displacement law (qualitative)?

λ_{ma x} T =

constant; zyada garam bodies choti wavelengths par peak karti hain.

Connections

Photoelectric Effect — Einstein ne $E = h ν$ ko light (photons) tak extend kiya.
Bohr Model of the Atom — quantized energy levels Planck ke quanta par build hote hain.
de Broglie Wavelength — wave–particle duality quantized energy se follow hoti hai.
Wien's Displacement Law & Stefan-Boltzmann Law — empirical facts jo Planck's law reproduce karta hai.
Boltzmann Distribution — derivation mein use hone wali $e^{- E / k T}$ weighting provide karta hai.