De Broglie hypothesis — matter waves λ = h - p: Question bank

2.3.5 · D5 · Physics › Modern Physics › De Broglie hypothesis — matter waves λ = h - p

Intuition Is bank ka use kaise karein

Ye woh questions hain jo unhe pakad lete hain jo "formula jaante hain" lekin idea ko feel nahi kiya. Har answer neeche ek chhoti chain of reasoning hai — reveal padho, phir apne dimag mein dobara derive karo. Agar koi claim obviously sahi lagta hai, to aksar wahi trap hoti hai. Master note: parent topic.

Key symbols jo tumhe padhne se pehle pata hone chahiye:

$λ$ = wavelength (associated wave ki crest-to-crest distance).
$v$ = particle ki speed (metres per second).
$m$ = particle ki mass; electron ke liye specifically $m_{e}$ likhte hain (electron rest mass, $9.11 \times 1 0^{- 31} kg$ ).
$p$ = momentum ("push"; slow bodies ke liye $p = m v$ ).
$h$ = Planck's constant ( $6.626 \times 1 0^{- 34} J\cdots$ ).
$K$ = kinetic energy.
$q$ = particle ka electric charge; elementary charge $e = 1.6 \times 1 0^{- 19} C$ hai.
$V$ = accelerating voltage (volts mein); charge $q$ jo $V$ se guzarta hai usse $K = q V$ milti hai.
$c$ = speed of light ( $3.0 \times 1 0^{8} m/s$ ).
$k_{B}$ = Boltzmann's constant ( $1.38 \times 1 0^{- 23} J/K$ ); $T$ = absolute temperature (kelvin mein).

Agar inme se koi bhi fuzzy lage, pehle parent note pe wapas jao.

True or false — justify

Each line: a statement, then ::: the verdict with the reason. Cover the reason, guess T/F and why, then check.

A heavier particle at the same speed has a shorter matter wavelength.

True.

λ = h / (m v)

; fixed

v

pe, zyada

m

matlab zyada

p

, to chhota

λ

. Mass denominator mein hai.

A heavier particle at the same kinetic energy pe bhi shorter wavelength hoti hai.

True, lekin alag reason se:

λ = h / 2 m K

, to fixed

K

pe zyada

m

phir bhi bada

p = 2 m K

deta hai, hence chhota

λ

. "Same speed" aur "same

K

" dono cases mein lighter particle zyada wavy hota hai.

A photon ki koi rest mass nahi hoti, isliye de Broglie relation usp pe apply nahi ho sakta.

False. Master form

λ = h / p

hai, jisme mass ka koi zikr nahi. Photon ka finite momentum

p = E / c

hota hai, to

λ = h / p

perfectly kaam karta hai — ye to poori idea ka seed tha.

Agar particle rest mein hai, to uski de Broglie wavelength zero hai.

False. Rest pe

p = 0

, to

λ = h / p \to \infty

— wavelength infinite hoti hai, zero nahi. "Wave" jiska koi momentum nahi uski koi spatial oscillation nahi hoti, jo dikhata hai ki truly stationary particle ke liye yeh picture toot jaati hai.

Matter wave matlab particle ek smeared blob of stuff mein spread ho gaya.

False. Ye ek probability amplitude hai (Born): particle phir bhi ek localised dot ke roop mein detect hota hai. Wave batata hai dot kahan land karne ki probability hai, na ki matter khud jelly hai. Dekho Wave–particle duality.

Electron ki speed double karne se uski wavelength double ho jaati hai.

False — ye half ho jaati hai.

λ \propto 1/ p \propto 1/ v

(non-relativistic), to zyada speed matlab chhoti wave. Log "motion quantity ka zyada hona" ko "zyada wavelength" se confuse karte hain.

Cricket ball ki koi de Broglie wavelength nahi hoti.

False. Hoti hai — approximately

1 0^{- 34}

m. Lekin ye ball ya kisi bhi slit se itni absurdly chhoti hai ki koi diffraction kabhi observable nahi hoti, isliye ball purely classical dikhti hai.

λ = h / (m v)

har speed pe valid hai, light-speed ke paas bhi.

False.

p = m v

sirf non-relativistically valid hai.

c

ke paas,

p = γ m v

hota hai, to sahi relativistic

p

ke saath

λ = h / p

use karna padega. Mass form se yahan start karna galat (bahut lamba) wavelength deta hai.

Same wavelength wale do particles ka momentum same hona chahiye.

True.

λ = h / p

ek one-to-one link hai:

p = h / λ

. Same

λ \Rightarrow

same

p

— chahe mass, charge, ya particle kuch bhi ho.

λ = h / p

ko matter ke liye Newton's laws se rigorously derive kiya gaya tha.

False. Matter ke liye ye ek hypothesis hai (light ke saath symmetry se postulate kiya gaya), derivation nahi. Iska sach experiment ne decide kiya — Davisson–Germer experiment ne 1927 mein electron diffraction dekha.

Spot the error

Each line states a flawed reasoning; the reveal names the flaw and fixes it.

"Ek electron aur proton same voltage se accelerate kiye jaayein to unki wavelength same hogi, kyunki unhe same energy milti hai."

Error: same

K = q V

lekin alag mass.

λ = h / 2 m K

, isliye lighter electron ki wave zyada lambi hoti hai. Equal energy

\neq =

equal wavelength.

"Electron ki wave chhoti karni ho, to use slow karo."

Error: ulta hai. Chhota

λ

chahiye to bada

p

chahiye, yaani tez karo.

λ \propto 1/ v

, isliye slow karne se wave lambi ho jaati hai.

100 V

electron ke liye

λ = 12.27/ V = 0.123 \overset{˚}{A}

use karunga."

Error: formula

λ = 12.27/ V

hai, na ki

12.27/ V

. Sahi answer

12.27/ 100 = 1.227 \overset{˚}{A}

hai. Square root

p = 2 m_{e} e V

se aata hai.

"Neutron neutral hai, isliye na

λ = h / 2 m q V

aur na hi koi de Broglie relation usp pe apply hoga."

Category error: voltage form assume karta hai charged particle (

q \neq = 0

) jo electric field se accelerate ho, isliye genuinely neutral neutron pe apply nahi hoti. Lekin ye ek special-case formula ki limitation hai, de Broglie ki nahi —

λ = h / p

aur

λ = h / 2 m K

theek kaam karte hain, aur thermal neutrons crystals se

\sim 1 \overset{˚}{A}

pe diffract hote hain.

λ = h / p

mein

h

tiny hai, isliye saare matter wavelengths undetectably small hain."

Error:

h

tiny hona sirf aadhi baat hai —

p

bhi tiny ho sakta hai. Slow electron ke liye

p

itna small hota hai ki

λ \sim

atomic size ho jaata hai. Mayne

h / p

ka ratio rakhta hai, sirf

h

nahi.

λ = h / p

ka matlab hai jaise jaise

p \to 0

, wavelength bhi zero ho jaati hai."

Error:

p

denominator mein hai, isliye

p \to 0

λ \to \infty

hota hai, na ki

\to 0

. Kam momentum

=

lambi wave.

"Photons aur electrons dono

λ = h / p

maante hain, isliye same wavelength wale photon aur electron ki energy same hogi."

Error: same

λ \Rightarrow

same

p

, lekin energy alag-alag relate hoti hai. Photon:

E = p c

. Electron (slow):

K = p^{2} /2 m

. Same

p

bahut alag energies deta hai.

Why questions

Gehri "kyun", ek ya do sentences mein real reasoning.

Wave behaviour everyday objects ke liye kyun gayab ho jaata hai?

Unka momentum

p = m v

enormous hota hai, isliye

λ = h / p \sim 1 0^{- 34} m

— kisi bhi obstacle ya slit se chhota. Diffraction tab hi dikhta hai jab

λ

aperture size ke comparable ho, jo macroscopic bodies ke liye kabhi nahi hota.

De Broglie ne apne starting point ke roop mein photon ka

λ = h / p

kyun use kiya, koi matter formula kyun nahi?

Kyunki

λ = h / p

mein light ke liye unique koi bhi property nahi — na

c

, na "massless". Agar relation mein kuch light-specific nahi hai, to symmetry suggest karti hai ki ye sab matter pe bhi hold karna chahiye.

Hum momentum form

λ = h / p

ko mass form

λ = h / (m v)

se zyada prefer kyun karte hain?

Momentum form universal hai — ye massless photons (

p = E / c

) aur relativistic particles (

p = γ m v

) dono ke liye kaam karta hai. Mass form sirf slow-speed ka special case hai aur dono ke liye toot jaata hai.

Zyaatar exam problems mein

λ = h / 2 m K

λ = h / (m v)

se zyada safe kyun hai?

Problems usually energy ya voltage deti hain, speed nahi.

p = 2 m K

directly

K

se momentum deta hai, ek extra step se bachata hai aur

p = m v

ko high speed pe galat use karne ki temptation se bhi.

Electrons se atoms image kyun ho sakti hain lekin visible light unhe resolve nahi kar sakti?

100 V

electron ka

λ \approx 1 \overset{˚}{A}

hota hai, jo atomic spacing se match karta hai, isliye ye lattice se diffract karta hai; visible light ka

λ \sim 5000 \overset{˚}{A}

hota hai, jo atomic-scale detail resolve karne ke liye bahut coarse hai.

De Broglie idea naturally uncertainty principle ki taraf kyun le jaata hai?

Definite wavelength wali wave (

\Rightarrow

definite

p

) poore space mein spread hoti hai (

\Rightarrow

undefined position). Particle ko localise karne ke liye wavelengths ka mix chahiye, jo

p

ko blur karta hai. Wahi trade-off Heisenberg Uncertainty Principle hai.

Bohr orbit condition

2 π r = nλ

matter wave ke roop mein kyun sense deta hai?

Electron ki wave ko orbit ke around smoothly join karna hota hai; sirf complete wavelengths ki whole numbers hi bina destructive cancellation ke fit hoti hain, jo allowed (quantised) orbits deti hain. Dekho Bohr model.

Edge cases

Boundary aur degenerate scenarios jo formula invite karta hai.

Exactly zero momentum wala particle:

λ

kya hoga?

λ = h /0 \to \infty

. Physically wave picture degenerate ho jaati hai — infinitely long wave ki koi localisation nahi hoti, jo perfectly stationary (hence position-uncertain) particle ke saath consistent hai.

Relativistic electron jiska

K

m_{e} c^{2}

ke comparable ho:

λ

ke liye kaunsa formula?

λ = h / (m v)

aur na

h / 2 m K

λ = h / p

use karo jahan

p = \frac{1}{c} K^{2} + 2 K m_{e} c^{2}

ho. Ye relativistic identity

E^{2} = (p c)^{2} + (m_{e} c^{2})^{2}

se directly aata hai: total energy

E = K + m_{e} c^{2}

likho, substitute karo, expand karo

(K + m_{e} c^{2})^{2} = K^{2} + 2 K m_{e} c^{2} + (m_{e} c^{2})^{2}

(m_{e} c^{2})^{2}

terms cancel karo, aur

p

ke liye solve karo.

Neutral particle (neutron) — kya voltage formula

λ = 12.27/ V

kabhi apply ho sakta hai?

Nahi, do reasons se: woh formula electron-specific hai (usme

m_{e}

aur

e

bake in hain), aur neutral particle ka

q = 0

hota hai isliye use electrically accelerate hi nahi kar sakte. Iske badle thermal energy

K = \frac{3}{2} k_{B} T

use karo.

Do identical electrons, ek doosre se double speed pe — wavelengths compare karo.

λ \propto 1/ v

, isliye faster wale ki wavelength half hogi. Same particle, alag momentum, alag wave.

Accelerated electron ke liye

V \to \infty

hone par

λ

ka kya hoga (relativity ignore karke)?

λ = 12.27/ V \to 0

. Zyada energy matlab chhoti wave — lekin

λ

sach mein vanish hone se pehle, relativistic

p

le leta hai aur simple formula fail ho jaata hai.

Same energy

E

carry karne wale photon aur electron mein: kis ki wavelength zyada lambi hogi?

Photon ki. Photon:

p = E / c

. Slow electron:

p = 2 m E

, jo same

E

ke liye bahut bada hai. Bada

p \Rightarrow

chhota

λ

, isliye equal energy pe electron kam wavy hota hai.

Recall Yahan har trap ki one-line summary

Lagbhag har galti isse aati hai ki bhool jaate hain ki $λ = h / p$ mein momentum denominator mein hai: zyada push $\Rightarrow$ chhoti wave, zero push $\Rightarrow$ infinite wave, aur mass sirf $p$ ke through matter karta hai.

Connections

Wave–particle duality — in saari traps ke peeche ka umbrella idea.
Davisson–Germer experiment — kyun hypothesis maani jaati hai.
Heisenberg Uncertainty Principle — wave nature se directly nikalti hai.
Bohr model — standing matter waves jo orbits fix karti hain.
Photoelectric effect — jahan photon momentum $p = E / c$ aata hai.
Compton effect — photon momentum ka independent proof.