Visual walkthrough — Photon properties — E = hf, p = h - λ

2.3.3 · D2 · Physics › Modern Physics › Photon properties — E = hf, p = h - λ

Hum ek boxed result dhundh rahe hain: $p=\frac{h}{\lambda }.$ Lekin hum ise likhkar trust nahi karenge — hum ise earn karenge.

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Step 1 — "Momentum" ka matlab kya hai (the shove)

WHAT. Koi bhi formula aane se pehle, ek moving cheez ko wall se takrate hue imagine karo. Momentum hai "motion mein store kiya hua shove" — woh cheez jo transfer hoti hai jab tum kisi cheez se takraate ho.

WHY yahan se shuru karo. Log $p=mv$ ("mass times velocity") yaad karte hain aur sochte hain momentum hai hi mass. Aisa nahi hai. Momentum ek shove budget hai: ek moving object jo cheez se takrata hai use kitna hard push karta hai. Ek truck walking pace par aur ek bullet — dono alag alag reasons se bada shove dete hain. Yeh picture yaad rakhna — shove, not mass — yahi poori key hai baad mein light ko momentum dene ki.

PICTURE. Do objects ek wall se takrate hain. Red arrow ki length = wall kitna recoil karta hai. Woh red arrow hi $p$ hai.

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Step 2 — School formula $p=mv$ light ke liye kyun kaam nahi karta

WHAT. Jaana-pehchaana rule likho aur ek photon par test karo.

$p=\underset{\text{mass}}{\underbrace{m}}\underset{\text{velocity}}{\underbrace{v}}$

Har symbol: $m$ = kitna stuff hai, $v$ = kitna fast hai. Dono multiply karo, shove milega.

WHY yeh fail karta hai. Ek photon (light ka ek lump) ki zero rest mass hoti hai: $m_0=0$. Ise plug karo: $p=0\times c=0.$ Yeh predict karta hai ki light koi shove carry nahi karti. Lekin comet tails Sun se door point karti hain, aur solar sails ko push milti hai — light zaroor cheezein push karti hai (dekho Radiation Pressure). Toh $p=mv$ ek incomplete rule hai. Hume real wali chahiye.

PICTURE. Formula $p=mv$ ek low-speed ramp hai; light top speed $c$ par rehti hai jahan yeh ramp reality describe nahi karta. Red "actual" curve zero se upar rehta hai wahan bhi jahan black $mv$ line ek massless cheez ko kuch bhi nahi bana deti.

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Step 3 — Real rule: energy–momentum triangle

WHAT. Relativity, $p=mv$ ko ek aisi relation se replace karta hai jo teen quantities ko connect karta hai — total energy $E$, momentum $p$, aur rest mass $m_0$ (dekho Relativistic Energy-Momentum Relation):

$E^2=\underset{\text{motion part}}{\underbrace{(pc{)}^2}}+\underset{\text{rest-mass part}}{\underbrace{(m_0{c}^2{)}^2}}$

Term by term:

• $E$ = object ki total energy.
• $pc$ = us energy ka woh hissa jo motion mein stored hai ($c$ = speed of light, $3\times 10^8$ m/s, yahan sirf ek unit-fixing constant hai).
• $m_0{c}^2$ = energy ka woh hissa jo sirf exist karne mein locked hai (rest energy).

WHY yeh tool use karo. Hume kuch aisa chahiye jo momentum ko energy se connect kare bina mass se multiply kiye. Notice karo ki is equation ki exact shape Pythagorean theorem ${\text{hyp}}^2={\text{leg}}^2+{\text{leg}}^2$ jaisi hai. Yeh coincidence nahi hai — energy, momentum-energy, aur rest-energy ek right triangle banate hain. Pythagoras tool hai kyunki yeh teen quantities perpendicular sides ki tarah combine hoti hain.

PICTURE. Ek right triangle: horizontal leg $pc$ (motion), vertical leg $m_0{c}^2$ (rest), hypotenuse $E$ (total). Red hypotenuse woh hai jo hum measure kar sakte hain.

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Step 4 — Triangle collapse karo: rest mass ko zero set karo

WHAT. Triangle mein $m_0=0$ daalo.

$E^2=(pc{)}^2+\underset{=0}{\underbrace{(0\cdot c^2{)}^2}}=(pc{)}^2$

Positive square root lo (energy positive hoti hai): $E=pc⟹p=\frac{E}{c}.$

Ab har symbol: $E$ = photon ki energy, $c$ = light speed, $p$ = uska momentum.

WHY yeh crucial move hai. Vertical leg $m_0{c}^2$ ko khatam karna triangle ko flat karke uski base par le aata hai. Hypotenuse aur horizontal leg ek hi line ban jaate hain: $E=pc$. Toh light ke liye, uski saari energy motion-energy hai — isme kuch bhi "at rest" nahi hai. Isliye ek massless cheez ka phir bhi momentum hota hai: $p=E/c$, aur uski energy $E$ zero nahi hai.

PICTURE. Step 3 ka triangle, lekin top vertex niche base par aa jaata hai — vertical leg shrink hokar kuch nahi ho jaata, hypotenuse (red) horizontal leg par flat ho jaata hai.

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Step 5 — Photon ki energy $E=hf$ feed karo

WHAT. Hum pehle se jaante hain ki har photon ki energy ek fixed packet hoti hai jo uski frequency se set hoti hai (parent mein Planck's Law and Blackbody Radiation se build hui): $E=\underset{\text{Planck’s constant}}{\underbrace{h}}\underset{\text{frequency}}{\underbrace{f}}$

• $h=6.626\times 10^{-34}$ J·s — ek energy "chunk" ka size per unit frequency.
• $f$ = kitne wave cycles per second pass hote hain (light ka "colour").

Is $E$ ko $p=E/c$ mein substitute karo: $p=\frac{E}{c}=\frac{hf}{c}.$

WHY. $p=E/c$ kisi bhi massless cheez ke liye sach hai. Ise is photon ke baare mein banane ke liye hum uski energy ko specific value $hf$ se replace karte hain. Koi naya assumption nahi — sirf ek earned result doosre mein plug kar rahe hain.

PICTURE. Ek conveyor: "colour dial" $f$ energy machine $E=hf$ ko feed karta hai, jiska output momentum machine $p=E/c$ mein jaata hai. Red final momentum output highlight karta hai.

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Step 6 — $c=f\lambda$ use karke frequency ko wavelength se trade karo

WHAT. Ek wave ki speed uski frequency times wavelength ke equal hoti hai: $c=\underset{\text{cycles/sec}}{\underbrace{f}}\times \underset{\text{length/cycle}}{\underbrace{\lambda }}$

• $f$ = cycles per second, $\lambda$ = ek cycle ki length (metres per cycle).
• Multiply karo: (cycles/sec)$\times$(metres/cycle) = metres/sec = ek speed, aur light ke liye woh speed hamesha $c$ hoti hai.

Hume jo ratio chahiye use isolate karne ke liye rearrange karo: $c=f\lambda ⟹\frac{f}{c}=\frac{1}{\lambda }.$

Ab Step 5 ka momentum lo aur $f/c$ ko $1/\lambda$ se swap karo: $p=\frac{hf}{c}=h\cdot \frac{f}{c}=h\cdot \frac{1}{\lambda }.$

WHY yeh swap karo. Experiments mein hum usually wavelength $\lambda$ measure karte hain (diffraction grating se), raw frequency nahi. $\lambda$ mein rewrite karne se formula directly usable ho jaata hai. Bridge $c=f\lambda$ hi hai jo frequency-language aur wavelength-language ko ek doosre se baat karaata hai.

PICTURE. Ek wave. Uski horizontal repeat-length $\lambda$ (red) label ki gayi hai; yeh right side $c$ speed se slide karti hai; fixed dot ke paas se per second kitne crests pass hote hain woh $f$ hai. Teeno quantities $c=f\lambda$ ke roop mein lock hain.

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Step 7 — Result, aur yeh har wavelength ke baare mein kya kehta hai

WHAT. Chain ko collect karo: $\boxed{p=\frac{h}{\lambda }}$

• $h$ = fixed Planck constant ("chunk size").
• $\lambda$ = photon ki wavelength.
• $p$ = uska momentum.

WHY yeh believable hai. Shape padho: $p\propto 1/\lambda$. Short wavelength $\to$ bada momentum; long wavelength $\to$ chhota momentum. Extremes check karo ("all cases" jo contract demand karta hai):

Case	$\lambda$	$p=h/\lambda$	Meaning
Gamma ray	tiny	huge	hammer ki tarah lagta hai (Compton electrons ko kick karta hai — dekho Compton Effect)
Visible light	~500 nm	tiny	solar sails ko bahut halka push karta hai
Radio wave	huge	~0	almost koi shove nahi
$\lambda \to \infty$	infinite	$\to 0$	koi wave nahi, koi push nahi — consistent
$\lambda \to 0$	zero	$\to \infty$	ever-shorter light = ever-harder punch

Koi bhi wavelength nonsensical answer nahi deti — formula har jagah theek behave karta hai.

PICTURE. Teen photons waves ki tarah draw kiye gaye hain — short (red, bada arrow), medium, long (tiny arrow). Arrow ki length = momentum, jo $\lambda$ badhne ke saath shrink hoti hai.

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Ek picture mein summary

Har step, compressed: colour dial $f$ → energy $E=hf$ → relativity triangle collapse karo ($m_0=0$) $E=pc$ mein → $c$ se divide karo → $f/c$ ko $1/\lambda$ se swap karo → $p=h/\lambda$ par land karo.

Recall Feynman: poora walk plain words mein

Momentum bas shove hai — ek moving cheez jo cheez se takraati hai use kitna hard push karti hai. School kehta hai shove = mass × speed, lekin woh sirf slow-lane ka rule hai. Universe ka real rule ek right triangle hai: total energy lambi side hai, "motion-energy" aur "rest-energy" do choti sides hain. Ek photon pure light hai — iska koi rest side hi nahi, toh uska triangle ek straight line mein flat ho jaata hai aur total energy equals motion-energy: $E=pc$. $c$ se divide karo aur photon ka momentum bas uski energy light ki speed se baanti hui hai. Uski energy colour se set hoti hai ($E=hf$), aur colour aur wavelength ek hi cheez ke do naam hain ($c=f\lambda$), toh end mein shove simply Planck ka chunk wavelength se divide kiya hua hai: $p=h/\lambda$. Kuch bhi weigh nahi kiya — phir bhi wall hilti hai.

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Connections

• Relativistic Energy-Momentum Relation — woh triangle jo humne Steps 3–4 mein flatten kiya.
• Compton Effect — experiment jahan $p=h/\lambda$ literally electrons ko kick karta hai.
• Radiation Pressure — Step 1 ka macroscopic shove.
• de Broglie Wavelength — $p=h/\lambda$ ko matter ke liye ulta run karta hai.
• Wave-Particle Duality — kyun ek lump dono wave ($\lambda$) aur particle ($p$) hai.
• Photoelectric Effect · Planck's Law and Blackbody Radiation — jahan se $E=hf$ aaya.