2.4.19 · D1Blackbody radiation from statistical mechanics — Planck dist
Intuition Ek hi idea jo sab kuch drive karta haiEk garam box isliye glowing karta hai kyunki uske andar chhoti-chhoti standing light-waves bhari hain, aur har wave ek chhota energy-storage bin hai jise heat bhar-na chahti hai. Poora topic bas itna hai — count karo kitne bins hain har pitch pe, aur heat unhe kitna bhar sakti hai — aur surprise yeh hai ki high-pitch bins bharne se mana kar dete hain jab tak heat ek poora "coin" ki energy na de.
Is parent note ki ek bhi line padhne se pehle, tumhe lagbhag ek dozen symbols aur ideas ki pakad chahiye. Neeche, har ek ko tabhi introduce kiya gaya hai jab usse pehle ke concepts aa chuke hain. Koi bhi cheez use nahi hoti jab tak ban na jaye.
Sab kuch ek cubic cavity ke andar hota hai — ek hollow metal box jiska side length L Figure 1 box ko ek akele trapped wave ke saath dikhata hai.
L L Picture: ek shoebox. Kyun chahiye: box ka finite size hi light ko specific patterns mein force karta hai (agla section). Infinite box any wave allow karta; finite box choosy hai.
x , y , z x y z 0 L Picture: teen rulers jo box ke teen edges ke saath chipke hain, ek corner pe milte hue. Kyun chahiye: "wave walls pe zero honi chahiye" kehne ke liye pehle kahan walls hain yeh batana padega — walls x = 0 , L y = 0 , L z = 0 , L
Definition Wave — ek wiggle jo travel karta haiLight wave ek electric-aur-magnetic wiggle hai jo space mein move karti hai. Picture: rope pe ripples jab tum use hilate ho.
Ek pure wave ko do numbers describe karte hain:
λ λ Picture: rope ke do paas-paas humps ke beech ka gap. Kyun chahiye: yeh batata hai wave kitni "stretched out" hai.
ν ν Picture: ek jagah khade raho aur naak se guzarne wale humps count karo per second. Kyun chahiye: frequency wave ki "pitch" ya "colour" hai — high ν ν har frequency ki kitni light hoti hai.
Yeh dono speed of light se ek saath bandhe hain:
c c ≈ 3 × 1 0 8 Kyun chahiye: yeh wavelength aur frequency ko connect karta hai.
Agar ek wave do walls ke beech trapped hai, toh woh koi bhi wave nahi ho sakti. Use ek locked pattern mein "baith" jaana hoga jise standing wave kehte hain: dono walls pe fixed, beech mein sirf wiggle. Figure 2 pehle teen allowed patterns draw karta hai jaise guitar-string ke notes.
Intuition Kyun sirf kuch khaas waves fit hoti hainWall wave ko zero pe pin kar deti hai (ek wave wall ko khud nahi hila sakti). Figure 2 dekho: har curve dono walls pe zero hai, aur sirf whole numbers of half-humps fit hote hain. Ek half-hump fit hota hai, do hote hain, teen hote hain — lekin kabhi do-aur-thoda nahi. Yeh bilkul guitar string ke notes jaisa hai.
"Walls pe vanish karna zaroori hai" ko equation mein daalo. x sin ( k x x ) k x x = 0 Sirf extra demand yeh hai ki yeh door wali wall x = L
Woh whole number of half-wiggles woh jagah hai jahan integers count karna shuru hota hai:
n x , n y , n z 1 , 2 , 3 , … ek axis ke saath . Picture: 3D dots ki ek lattice, har dot ( n x , n y , n z ) Kyun chahiye: in dots ko count karna literally count karna hai ki box mein kitne light-modes hain. (Warning: combined radial number n
Ek mode = ek specific allowed standing-wave pattern (n x , n y , n z Picture: lattice mein ek dot. Topic ke pehle aadhe hisse ka poora sawaal yahi hai: har frequency pe kitne modes hain?
Teen integers juggle karne ki jagah, physicists wave ki direction aur tightness ko ek single arrow mein bundle karte hain:
k ∣ k ∣ k ∣ k ∣ = 2 π / λ tightly packed hain (bada ∣ k ∣ k x , k y , k z Picture: ek arrow jiski length badhti hai jab wave bluer hoti jaati hai.
Modes count karne ke liye hum har allowed pattern ko ek dot ( n x , n y , n z ) Figure 3 yeh lattice aur use count karne wala octant-sphere dikhata hai.
n n ≡ n x 2 + n y 2 + n z 2 ( n x , n y , n z ) Dhyan do: yeh n n x combined distance hai. Picture (Figure 3): corner se dot tak arrow ki length. Kyun chahiye: same wave-tightness ∣ k ∣ n
Intuition Kyun sphere ka ek-aathwan hissaKyunki teeno integers positive hain, isliye har dot uss corner mein rehta hai jahan n x , n y , n z > 0 aath corners mein se ek hai (ek octant). Radial distance n n 8 1 × ( ball volume ) N ( n ) = 8 1 ⋅ 3 4 π n 3 = 6 π n 3
N ( n ) N ( n ) n Picture: us eighth-ball ke andar kitne dots hain. Kyun chahiye: frequency ke saath iski rate of change exactly density of states g ( ν )
g ( ν )
d ν d d ν Kyun chahiye: spectrum slice by slice define hota hai.
d ν d d ν d [ ⋅ ] jaise ν Yeh tool kyun, koi aur nahi: N ( ν ) ν total modes hain; ek single slice mein modes paane ke liye growth rate chahiye — exactly wahi ek derivative compute karta hai. Picture: N ν
Definition Polarization — 2 ka factorHar wavevector k do independent light-wiggle orientations hote hain (kaho horizontal aur vertical), jinhein polarizations kehte hain. Picture: wahi arrow, lekin field side-to-side ya up-down hil sakta hai. Kyun chahiye: mode count double karna isi liye hai ki g ( ν )
Ab hum g ( ν ) build kar sakte hain. Pehle N ∣ k ∣ = nπ / L = 2 π ν / c n = 2 Lν / c N ( ν ) = 6 π ( c 2 Lν ) 3 = 3 c 3 4 π L 3 ν 3
V = L 3 V V = L 3 Kyun chahiye: N ( ν ) poore box ke modes count karta hai, lekin hum per unit volume count chahte hain taki answer box ki size pe depend na kare. V = L 3 L 3
g ( ν ) g ( ν ) d ν ν ν + d ν Picture: is pitch pe kitne bins tuned hain. Iski ν 2 bahut zyada hain.
Ab hum puchte hain ki har bin kitna bhar jaata hai. Yeh thermodynamics hai.
T k B T k B ≈ 1.38 × 1 0 − 23 Picture: k B T Kyun chahiye: k B T Equipartition theorem .
β = k B T 1 ⇒ β ⇒ β Kyun chahiye: yeh statistical-mechanics ke formulas ko cleaner banata hai; β
e x e ≈ 2.718 e x x apna khud ka rate of change hai. Yeh tool kyun: thermal physics mein probabilities e − energy / k B T Boltzmann factor ) — costly states exponentially rare hote hain. Yahi exponential baad mein ν 2 Picture: ek curve jo zero ki taraf kisi bhi power se zyada tezi se gir jaata hai jab x
h h ν h ≈ 6.63 × 1 0 − 34 ν h ν coins mein energy gain ya lose kar sakta hai — kabhi fraction nahi. Picture: ek vending machine jo sirf exact coins leta hai; ek ν h ν Kyun chahiye: yeh akela rule Planck ka poora fix hai. High-ν k B T
ϵ n = nh ν ϵ 0 , h ν , 2 h ν , … n = 0 , 1 , 2 , … coins ki sankhya hai — letter n Kyun chahiye: yeh discrete rungs (continuous ramp nahi) woh cheez hai jo classical equipartition todti hai.
∑ ∑ n = 0 ∞ n = 0 n Picture: har possibility ko ek total mein stack karna.
Z Z = ∑ n e − β ϵ n Picture: "har state kitni reachable hai" ka ek grand tally. Kyun chahiye: yeh master bookkeeper hai — ek baar Z Partition function .
For our coin-ladder ϵ n = nh ν x = β h ν e − x < 1 Z = ∑ n = 0 ∞ e − n β h ν = ∑ n = 0 ∞ ( e − x ) n = 1 − e − x 1
⟨ ϵ ⟩ ⟨ ⟩ ⟨ ϵ ⟩ T on average kitna bhara hai. Kyun chahiye: yeh recipe ka "energy quantize karo" wala aadha hissa hai. g ( ν )
Figure 4 punchline dikhata hai: g ( ν ) ν 2 ⟨ ϵ ⟩
Intuition Poora topic ek product meinis pitch pe energy u ( ν ) d ν = kitne bins hain g ( ν ) × har bin kitna bhara hai ⟨ ϵ ⟩ × d ν Figure 4 compare karo: dashed curve upar, dotted curve neeche, red product beech mein.
u ( ν , T ) u ( ν , T ) d ν [ ν , ν + d ν ] T Picture: har colour pe glow-curve ki height (Figure 4 mein red curve). Yahi hai Planck distribution — parent note ke end mein trophy.
Standing waves fit the box
Wave: lambda and nu, c = lambda nu
Count dots in one octant N of n
Density of states g of nu
Divide by volume V = L cubed
Boltzmann factor e to the minus beta E
Planck distribution u of nu T
Cover the right side and answer aloud; reveal to check.
Box ki walls coordinates mein kahan hain? x = 0 , L y = 0 , L z = 0 , L
ν Frequency — wiggles per second, hertz mein (1/s); yeh wave ka colour/pitch hai.
c λ ν c = λ ν ν = c / λ
Boundary condition k x = n x π / L Wave sin ( k x x ) x = L sin π k x L = n x π
Ek "mode" kya hota hai? Ek specific allowed standing-wave pattern (ek
k aur ek polarization).
Yahan use hone wale n Per-axis
n x , n y , n z = ek axis ke saath half-wiggles; radial
n = n x 2 + n y 2 + n z 2 = dot tak distance.
Hum modes ko sphere ke ek-aathwen hisse mein kyun count karte hain? Kyunki n x , n y , n z
N ( ν ) g ( ν ) ×2 (polarizations), ν V = L 3
Density of states likho. g ( ν ) = 8 π ν 2 / c 3
Boltzmann factor e − energy / k B T Ek state ki relative likelihood; costly states exponentially rare hote hain.
"Energy coin" h ν Woh sabse chhota energy lump jo frequency ν
⟨ ϵ ⟩ = − ∂ ln Z / ∂ β Woh derivative automatically weighted average energy ∑ ϵ n e − β ϵ n / ∑ e − β ϵ n
Quantized mode ke liye ⟨ ϵ ⟩ ⟨ ϵ ⟩ = h ν / ( e h ν / k B T − 1 )
Spectrum ke liye master product likho. u ( ν ) d ν = g ( ν ) ⟨ ϵ ⟩ d ν
Prerequisite vault links: Density of states · Partition function · Equipartition theorem · Bose-Einstein statistics · Rayleigh-Jeans law · Wien's law · Stefan-Boltzmann law · parent pe wapas jaao → 2.4.19 Blackbody radiation from statistical mechanics — Planck distribution (Hinglish)
