De-Broglie hypothesis

De-Broglie hypothesis

1. Hypothesis — Matter Has Wavelength

In 1924 Louis de Broglie proposed that every moving particle with momentum $p$ has an associated matter wave with wavelength:

and frequency $ν = E/h$, where $E$ total energy. This symmetrically extends $E=hν, p=h/λ$ from photons to matter (photon $m=0$ was wave-particle; now particle is also wave).

2. Why Plausible

• Bohr quantization $mvr = nħ$ is equivalent to standing wave condition

with $λ=h/p$ — De Broglie explains Bohr ad hoc as electron wave closing on itself.

• Relativity: if photon $p=h/λ$, why not electron?

3. Experimental Proof — Davisson-Germer (1927)

• Accelerated electrons ($E=eV$) have $λ = h/√(2m eV) ≈12.27/√V$ Å (with $V$ in volts: $λ(Å)=12.27/√V$).
• Davisson-Germer: 54 eV electrons ($λ≈1.67$ Å) scattered from Ni crystal at $θ=50°$ showed strong peak at Bragg condition $2d sinθ = nλ$, matching $λ$ predicted.
• G.P. Thomson: electron beam through thin foil gave Debye-Scherrer rings (like X-ray diffraction) — electrons diffract.

Thus electrons (and neutrons, atoms) diffract as waves.

4. Worked Examples

(a) Electron accelerated 100 V: λ=12.27/√100=1.227 Å (comparable to lattice spacing, so diffracts)

(b) Electron 1 eV: λ≈12.27 Å (X-ray region)

(c) Ball m=0.1 kg, v=1 m/s: λ=6.6e-34/0.1=6.6e-33 m absurdly small ⇒ not observed, classical.

(d) Thermal neutron (E≈0.025 eV): λ≈1.82 Å (useful for crystal diffraction)

5. Group vs Phase Velocity (mention)

Phase $v_p = E/p = ω/k$, group $v_g = dω/dk = p/m = v$ (particle velocity). For free particle, $v_p =c²/v$ >c? Actually $v_p$ can exceed $c$ but no information.

6. Exam Points

• State $λ=h/p$, $λ=12.27/√V$ Å — 5 marks, very frequent (almost certain to ask formula).
• Explain Bohr $2πr=nλ$ derivation — 5 marks, frequent.
• Describe Davisson-Germer experiment with $λ$ and Bragg — 7 marks, very frequent (common proof question).
• Compute $λ$ for eV, ball, neutron — 3-5 marks, frequent.