Foundations — Photon properties — E = hf, p = h - λ

2.3.3 · D1 · Physics › Modern Physics › Photon properties — E = hf, p = h - λ

Isse pehle ki tum $E=hf$ ya $p=h/\lambda$ padh sako, tumhe yeh jaanna zaroori hai ki wave kya hoti hai, "frequency" aur "wavelength" pictures mein kya matlab rakhte hain, speed $c$ unhe kaise jorti hai, aur "momentum" aur "rest mass" actually kya hote hain. Hum inhe ek ek karke build karte hain — har naya symbol apni jagah earn karta hai pehle, phir agla aata hai.

1. Ek wave, ek picture ki tarah

Neeche figure mein upar wali curve dekho. Yeh ek snapshot hai — ek frozen instant. Curve ek crest tak uthti hai, ek trough tak girti hai, phir uthti hai. Yeh repeating shape hi ek single cheez hai jo is page par baaki sab kuch define karne ke liye chahiye.

• Ek crest ki height ko amplitude kehte hain — wiggle kitni tall hai. Roz ke shabdon mein, badi amplitude = zyada bright light.
• Ek crest se bilkul agle crest tak ki horizontal distance ko wavelength kehte hain.

Hum dono ki zaroorat padegi, lekin photons ke liye surprise (jo parent note mein build hota hai) yeh hai ki amplitude ek single photon ki energy set nahi karta — wavelength karta hai.

2. Wavelength — symbol $\lambda$

Hume yeh kyun chahiye? Kyunki light ke liye "color" hi wavelength hai. Red light ki $\lambda$ lambi hoti hai (lagbhag $700$ nm), blue light ki $\lambda$ chhoti hoti hai (lagbhag $450$ nm). Jab parent note $p=h/\lambda$ likhta hai, toh woh keh raha hai ki momentum is crest-to-crest distance se judi hai.

3. Frequency — symbol $f$

Hume usi wave ke doosre description ki zaroorat kyun hai? Kyunki $\lambda$ space mein ek picture hai (ek snapshot), jabki $f$ time mein ek picture hai (ek jagah khade raho aur crests count karo jo guzar rahi hain). Dono alag viewpoints se usi wave ko describe karte hain.

Figure mein, cyan dot ek fixed jagah par baitha hai jabki wave uske paas se stream karti hai. Jab bhi ek crest dot tak pahunchti hai, hum counter mein ek tick karte hain. Ek second mein ticks ki sankhya $f$ hai. Tez wiggling (crests aas paas, chhota $\lambda$) → har second zyada ticks → bada $f$. Yeh tumhara pehla clue hai ki $\lambda$ aur $f$ inversely linked hain — agla section ise exact banata hai.

4. Bridge — speed $c$ aur $c=f\lambda$

Ab woh key relation jo ruler ($\lambda$) ko clock ($f$) se jorti hai:

YEH SACH KYUN HAI? Ek second mein, $f$ crests tumhare paas se guzarti hain (yahi $f$ ka matlab hai). Har crest $\lambda$ metres lambi hai. Toh ek second mein jo total wave length guzri woh hai (crests ki sankhya) $\times$ (har ek ki length) $=f\times \lambda$. Lekin "length passing per second" exactly speed hai. Isliye $c=f\lambda$.

5. Planck's constant $h$ — exchange rate

Topic ko yeh kyun chahiye? $f$ cycles-per-second mein measure hoti hai; energy joules mein measure hoti hai. Ek ko doosre mein convert karne ke liye tumhe sahi units wala ek conversion factor chahiye. Woh factor $h$ hai. $E=hf$ mein, $h$ literally price tag hai: frequency ka har hertz ek $h$ energy ka kharcha karta hai.

6. Energy $E$ aur electronvolt

Doosra energy unit kyun introduce karein? Ek single photon kuch aisa $10^{-19}\text{J}$ carry karta hai — padhne ke liye ek bura number. $1.6\times 10^{-19}$ se divide karo aur tumhe friendly kuch "electronvolts" milte hain. Ek green photon lagbhag $2.5\text{eV}$ ka hota hai; yeh ek aisa number hai jo insaan dimag mein rakh sakta hai.

7. Momentum $p$ — relativity se pehle

Hum $p$ se yahan milte hain taaki parent note ke $p=h/\lambda$ ko ek ghar mile. Lekin ek trap hai: everyday formula $p=mv$ kehta hai ki massless cheez ($m=0$) ka momentum zero hota hai. Light ki rest mass zero hai phir bhi clearly woh cheezein push karti hai (comet tails, solar sails). Toh $p=mv$ poori kahani nahi ho sakti — hume deeper version chahiye, jo aage aa raha hai.

8. Rest mass $m_0$ aur relativistic upgrade

Einstein ki relativity schoolbook wali $p=mv$ ko energy, momentum, aur rest mass ko connect karne wale ek fuller law se replace karti hai:

Tum ise yahan derive nahi karte (yeh Relativistic Energy-Momentum Relation se aata hai) — tumhe bas ise padhna hai. Yeh kehta hai: kisi cheez ki energy do pieces se bani hai, ek motion piece $(pc)$ aur ek rest piece $(m_0{c}^2)$, jo ek right triangle ki do legs ki tarah combine hote hain.

Figure mein hypotenuse $E$ hai, ek leg $pc$ hai, doosri leg $m_0{c}^2$ hai. Photon ke liye rest-mass leg zero tak shrink ho jaati hai, toh triangle flat collapse ho jaata hai aur hypotenuse baaki bachi leg ke equal ho jaati hai: $m_0=0⟹E^2=(pc{)}^2⟹E=pc⟹p=\frac{E}{c}.$

Woh single collapse hi parent note par $p=h/\lambda$ ka engine hai. Ab $p=E/c$ zero mass ke saath bhi non-zero momentum deta hai — everyday formula ka trap dodge ho jaata hai.

Har symbol kaise do formulas ko feed karta hai

Ise top-to-bottom padho: wave ki pictures tumhe $\lambda$ aur $f$ deti hain; speed $c$ unhe bridge karti hai; $h$, $f$ ko energy $E$ mein badalta hai; $m_0=0$ wala relativistic triangle energy ko momentum $p$ mein badalta hai. Har arrow is page ka ek justified step hai.

Equipment checklist

Khud ko test karo — right side cover karo aur reveal karne se pehle answer do.

Connections

• Parent topic — Photon properties — jahan yeh symbols $E=hf$ aur $p=h/\lambda$ mein combine hote hain.
• Relativistic Energy-Momentum Relation — §8 mein use hue energy–momentum triangle ka poora justification.
• Planck's Law and Blackbody Radiation — jahan $h$ aur energy chunks ka idea paida hua.
• Photoelectric Effect — woh experiment jisne "energy $f$ par depend karti hai, amplitude par nahi" force kiya.
• de Broglie Wavelength — matter ke liye $p=h/\lambda$ reuse karta hai.
• Wave-Particle Duality — woh big picture jiske liye yeh foundations serve karte hain.