De Broglie hypothesis — matter waves λ = h - p

2.3.5 · Physics › Modern Physics

1. Matter wave kya hota hai?

YEH kya nahi hai: yeh particle ka physically ordinary space mein wave mein phail jaana nahi hai. Yeh ek probability ki wave hai — jahaan wave intense hai, wahaan particle milne ki probability zyada hai. Lekin $λ$ nikaalte waqt, isse simple wavelength ki tarah treat karo.

2. $λ = h / p$ ko first principles se derive karna

Hum ise photon ko steel-man karke aur symmetry demand karke banate hain.

Useful rewrites (derive karo, memorize mat karo)

Mass $m$ wala ek particle jiska kinetic energy $K$ hai: $K = \frac{1}{2} m v^{2} = \frac{p ^{2}}{2 m} \Rightarrow p = 2 m K .$ Kyun? $p = m v$ toh $p^{2} = m^{2} v^{2} = 2 m (\frac{1}{2} m v^{2}) = 2 m K$ . Isliye $λ = \frac{h}{2 m K} .$

Charge $q$ wala particle jo potential $V$ se accelerate hota hai, use $K = q V$ milta hai: $λ = \frac{h}{2 m q V} .$ Electron ke liye, numbers plug karo (ek baar karo, phir reuse karo): $λ = \frac{12.27}{V} \overset{˚}{A} (V in volts) .$ Yeh constant kyun? $\frac{h}{2 m _{e} e} = \frac{6.626 \times 1 0 ^{- 34}}{2 ( 9.11 \times 1 0 ^{- 31} ) ( 1.6 \times 1 0 ^{- 19} )} \approx 1.227 \times 1 0^{- 9} m\cdotV^{1/2} = 12.27 \overset{˚}{A} \cdotV^{1/2} .$

Figure — De Broglie hypothesis — matter waves λ = h - p

3. Worked examples

4. Common mistakes (Steel-man + fix)

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho tum ek ball phenko. De Broglie ne kaha ki ball ke saath secretly ek tiny ripple aati hai, jaise ek wave. Yeh ripple kitni badi hai? Yeh depend karta hai ki ball ka kitna "push" (momentum) hai — zyada push, choti ripple. Real ball ke liye ripple itni unbelievably tiny hai ( $1 0^{- 34}$ m!) ki tumhe kabhi notice nahi hogi. Lekin ek teeny electron ke liye, ripple ek atom ke size ki hoti hai — itni badi ki electrons phail kar wave patterns bana sakte hain, bilkul paani ki waves ki tarah gaps se. Toh matter secretly thoda wave-y hota hai; hum bas tab dekhte hain jab cheezein super small hoti hain.

5. Forecast-then-Verify

Flashcards

De Broglie relation (state karo)

λ = h / p

, momentum

p

wale particle ka wavelength;

h

= Planck's constant.

Cricket ball ki wave nature kyun kabhi nahi dikhti?

Uska momentum bahut bada hota hai, toh

λ = h / p \sim 1 0^{- 34}

m — kisi bhi slit/object se bahut chota, isliye koi diffraction nahi.

λ

ko kinetic energy

K

aur mass

m

ke terms mein express karo

λ = h / 2 m K

(kyunki

p = 2 m K

V

volts se accelerate hua electron:

λ

ke liye quick formula?

λ = 12.27/ V

Å.

Equal kinetic energy par, kaun zyada wavy hai, electron ya proton, aur kis factor se?

Electron;

λ \propto 1/ m

, factor

m_{p} / m_{e} = 1836 \approx 42.8

Konse experiment ne matter waves confirm kiye?

Davisson–Germer (1927), nickel crystal se electron diffraction.

Kya

λ = h / p

matter ke liye ek derivation hai ya hypothesis?

Ek hypothesis hai (light ke saath symmetry se postulate); experiment se confirm hua.

Non-relativistic particle ki speed double karne se

λ

ka kya hoga?

Half ho jaata hai (

λ \propto 1/ p \propto 1/ v

Matter wave actually kya represent karta hai (Born)?

Ek probability amplitude — jahaan particle milne ki probability hai, koi physical smear nahi.

Connections

Photoelectric effect — photons ko momentum $p = E / c$ deta hai, $λ = h / p$ ka beej.
Davisson–Germer experiment — electron waves ka experimental proof.
Heisenberg Uncertainty Principle — wave nature se aata hai ( $Δ x Δ p ≳ ℏ$ ).
Bohr model — quantized orbits = standing matter waves ( $2 π r = nλ$ ).
Wave–particle duality — umbrella concept.
Compton effect — photon momentum confirm hua.