Foundations — Compton scattering — wavelength shift derivation

2.3.4 · D1 · Physics › Modern Physics › Compton scattering — wavelength shift derivation

Yeh page assume karta hai ki aapne parent note mein se koi bhi symbol pehle nahi dekha. Hum har ek symbol ko ekdum ground up se, ek aisi order mein build karenge jahan har ek cheez pichli par tikhi ho. Ant tak aap parent derivation ko bina kisi letter par rukhe padh paoge.

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0. Woh scene jo hum describe kar rahe hain

Figure dekho. Ek photon (lavender arrow) left se aakar ek electron (mint dot) se takraata hai jo still baitha hai. Collision ke baad do cheezein bahar nikalti hain: photon (ab coral, ek angle $\theta$ par mod gaya) aur electron (slate arrow, dusri taraf angle $\varphi$ par mod gaya). Neeche sab kuch bas is picture ke pieces ko naam dena hai.

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1. Wavelength $\lambda$ — "ek ripple ki length"

Picture: samundar ki wiggly line imagine karo. Ek wave-top se agli wave-top ka gap $\lambda$ hai. Neeche ke figure ki top strip dekho — tight ripple chhota $\lambda$ hai, loose ripple bada $\lambda$ hai.

Topic ko yeh kyun chahiye: Compton scattering ka poora result yahi hai ki bounce ke baad wavelength badhti hai. "Pehle wali" wavelength ko $\lambda$ aur "baad wali" ko ${\lambda }^{\prime }$ likhte hain ("lambda-prime" padho — woh choti tick ka matlab bas nayi wali hai). Jis quantity ki sabko chinta hai woh hai increase:

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2. Frequency $f$ aur light-speed link $c=f\lambda$

Light hamesha ek hi speed $c\approx 3\times 10^8$ m/s par travel karti hai (speed of light). Ek second mein wave $c$ distance aage jaati hai; agar $f$ crests guzarti hain aur har ek $\lambda$ lambi hai, toh $c=f\lambda$. Toh frequency aur wavelength ek hi baat kehne ke do tarike hain:

$f=\frac{c}{\lambda }$

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3. Planck's constant $h$ aur photon energy $E=hf$

Revolutionary idea (from the Photoelectric effect): frequency $f$ ki light sirf poore packets mein hi di ja sakti hai, har ek mein energy hoti hai

$E=hf=\frac{hc}{\lambda }$

Last step mein bas $f$ ko $c/\lambda$ se replace kiya. Toh choti wavelength matlab badi energy — isliye X-rays use hoti hain, lamplight nahi.

Topic ko yeh kyun chahiye: parent note mein conservation of energy poori tarah $hc/\lambda$ se likhi hai. Dekho Photon momentum and energy.

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4. Momentum $p$ — "push" jo ek chalti cheez carry karti hai

Picture: ek rolling bowling ball mein bahut momentum hota hai (heavy, moving); ek drifting feather mein almost kuch nahi. Collision mein momentum ke arrow-sum ka total pehle aur baad mein barabar rehta hai (dekho Conservation of momentum).

Photon ke liye koi mass nahi, phir bhi yeh push carry karta hai. Sahi rule (aage justify kiya gaya) hai:

Dobara notice karo: chota $\lambda$ → bada push. Photon ek choti ball hai jisme energy $hc/\lambda$ aur momentum $h/\lambda$ hoti hai.

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5. Photon ke liye $E=pc$ kyun — relativistic energy relation

Ordinary mechanics kehti hai $E=\frac{1}{2}m{v}^2$, lekin photon ka koi mass nahi aur yeh $c$ par move karta hai, toh woh formula yahaan bekaar hai. Einstein ka replacement (dekho Relativistic energy-momentum relation) kisi bhi object ke liye master equation hai:

Ise ek right triangle ki tarah padho (figure dekho): total energy $E$ lambi side (hypotenuse) hai, aur do choti sides hain "motion part" $pc$ aur "existence part" $m{c}^2$. Pythagoras inhe ek saath jodta hai.

• Photon ($m=0$): "existence" side gayab ho jaati hai, toh $E=pc$, yaani $p=E/c$. Yahi Section 4 ka box hai.
• Electron ($m=m_e\ne 0$): dono sides survive karti hain, toh $E_e=\sqrt{(p_{e}c{)}^2+(m_e{c}^2{)}^2}$. Isliye electron ko relativistically treat karna padta hai — uski recoil tez ho sakti hai, aur sirf yahi triangle correct rehta hai.

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6. Electron mass $m_e$ aur uski rest energy $m_e{c}^2$

Topic ko yeh kyun chahiye: final shift formula $m_{e}c$ se divide hota hai. Poore atom ka mass use karna (ek common trap) shift ko almost zero kar deta — yeh actually un extra "unshifted peak" ko explain karta hai jo Compton ne dekha, lekin moving-electron peak ke liye $m_e$ chahiye.

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7. Angles $\theta$ aur $\varphi$

Picture: Section 0 ke figure mein, $\theta$ incoming direction aur coral (outgoing photon) arrow ke beech khulta hai; $\varphi$ line ke neeche electron ke slate arrow ki taraf khulta hai. Hum inhe alag rakhte hain kyunki dono objects alag alag direction mein jaate hain.

$\varphi$ final answer mein kyun nahi aata: hum sirf photon ko measure karte hain. Parent derivation do momentum equations ko square-and-add karta hai taaki ${\cos }^2\varphi +{\sin }^2\varphi =1$ se $\varphi$ gayab ho jaye.

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8. Cosine, aur woh special angles jo hume chahiye honge

$\cos \theta$ (cosine) jawab deta hai: outgoing arrow ka kitna hissa abhi bhi aage ki taraf point karta hai? $\theta =0$ par photon muda nahi, toh woh poori tarah aage point karta hai, $\cos 0=1$. Sideways muda, $\cos 90^{\circ }=0$. Poori tarah waapis muda, $\cos 180^{\circ }=-1$.

Topic ko bas itni hi trigonometry chahiye, aur yeh teen headline cases ko ek nazar mein explain karta hai:

angle $\theta$	$\cos \theta$	$1-\cos \theta$	matlab
$0^{\circ }$	$1$	$0$	koi turn nahi → koi shift nahi
$90^{\circ }$	$0$	$1$	sideways → shift $={\lambda }_C$
$180^{\circ }$	$-1$	$2$	backscatter → maximum shift $=2{\lambda }_C$

Beech ke har case smoothly in dono ke beech aata hai, kyunki $\cos \theta$ smoothly $1$ se $-1$ tak slide karta hai.

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9. Vocabulary ko ek saath rakhna

Ab parent ke final box mein har symbol define ho chuka hai:

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Prerequisite map

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Equipment checklist

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Connections

• Photon momentum and energy — jahan $E=hf$ aur $p=h/\lambda$ aate hain.
• Relativistic energy-momentum relation — triangle $E^2=(pc{)}^2+(m{c}^2{)}^2$.
• de Broglie wavelength — ulta idea: matter ki bhi wavelength hoti hai.
• Conservation of energy aur Conservation of momentum — pool-table rules.
• Photoelectric effect — pehla proof ki light packets mein aati hai.
• X-ray production and Bremsstrahlung — jahan incoming X-rays banti hain.
• Parent: Compton scattering derivation.