De Broglie Theory, Matter Waves & Associated Quantum Concepts

De Broglie Hypothesis & Origin of Matter Waves

• Louis de Broglie (1924) postulated that every material particle of momentum $p$ possesses an associated wave of wavelength $\lambda$.
  • Fundamental relation: $\lambda = \frac{h}{p}$ where $h$ is Planck’s constant.
• Connects two classical pictures:
  • Particle‐like: definite mass $m$, velocity $v$, momentum $p = mv$.
  • Wave‐like: frequency $\nu$, wavelength $\lambda$, phase $\phi$.
• For photons the duality was already known ( $E = h\nu = pc$ ); de Broglie generalized it to “matter”.
• Significance: provides bridge from classical mechanics to quantum mechanics, explaining quantised orbits (Bohr), diffraction of electrons, etc.

Key Mathematical Relations for $\lambda$

• Non-relativistic particle with kinetic energy $E_{\text{K}}$:
  • $\lambda = \frac{h}{\sqrt{2mE_{\text{K}}}}$.
• Thermal particle in 3-D gas (equipartition ⇒ $E_{\text{K}} = \tfrac{3}{2}kT$):
  • $\lambda = \frac{h}{\sqrt{3mkT}}$.
• Electron accelerated through potential difference $V$:
  • $E_{\text{K}} = eV \Rightarrow \lambda = \frac{h}{\sqrt{2meV}} = \frac{12.26\;\text{Å}}{\sqrt{V(\text{volts})}}$.
  • Example: $V = 54\,\text{V} \Rightarrow \lambda \approx 1.67\,\text{Å}$ (matches Ni crystal spacing ⇒ strong diffraction peak).
• Extreme relativistic limit ( $v \rightarrow c$ ) links to Einstein relation $E = mc^2$.

Character & Terminology of Matter Waves

• Not electromagnetic; amplitude, phase, frequency are probability related (Born / Schrödinger interpretation).
• Described by wave function $\psi(x,t) = A e^{i( kx - \omega t)}$.
• Important descriptors:
  • $A$: amplitude (probability density envelope).
  • $k = \tfrac{2\pi}{\lambda}$: wave-number.
  • $\omega = 2\pi\nu$: angular frequency.
• For an isolated particle the measurable velocity is identified with the group velocity of its matter wave packet.

Phase Velocity ($V<em>p$) vs Group Velocity ($V</em>g$)

• Single monochromatic component: $V_p = \frac{\omega}{k}$.
• Build realistic packet by superposition of two close components:
  • $y<em>1 = A\sin(\omega</em>1 t - k<em>1 x)$, $y</em>2 = A\sin(\omega<em>2 t - k</em>2 x)$.
  • Resultant:
    $y = 2A\cos\left(\frac{\Delta \omega}{2}t - \frac{\Delta k}{2}x \right) \sin\left(\bar{\omega} t - \bar{k} x \right)$
    where $\Delta \omega = \omega<em>2-\omega</em>1$, $\bar{\omega}=\tfrac{\omega<em>1+\omega</em>2}{2}$ (similarly for $k$).
• Envelope (group) travels with
  $V_g = \frac{\Delta \omega}{\Delta k} \;\xrightarrow[]{\Delta \to 0}\; \frac{d\omega}{dk}$.
• For matter waves governed by free‐particle dispersion $\omega = \tfrac{\hbar k^2}{2m}$ ⇒ $V_g = \frac{\hbar k}{m}=v$ (the actual particle speed).
• Phase velocity: $V_p = \tfrac{\omega}{k}= \tfrac{v}{2}$ in non-relativistic theory; always < c for material particles.

Construction of a Wave Packet

• Many components superpose → constructive interference inside a narrow region, destructive outside ⇒ localized packet.
• Amplitude of packet varies periodically; distance between maxima of the envelope defines the packet width.
• Attests need for range of $k$ values ⇒ intimately linked to Heisenberg uncertainty.

Experimental Verification – Davisson–Germer Experiment

• Setup:
  • Heated Ni crystal (monocrystalline) as target.
  • Electron gun fed from high-tension battery (~20–70 V).
  • Rotatable, double-valved Faraday cylinder detector inside high vacuum avoids secondary–electron contamination.
• Procedure:
  1. Accelerate electrons by chosen $V$.
  2. Measure intensity of electrons elastically scattered at angle $\theta$.
  3. Plot intensity vs $\theta$ for different $V$.
• Observation:
  • Pronounced intensity peak at $\theta \approx 50^\circ$ when $V = 54\, \text{V}$.
  • Compute $\lambda$ by de Broglie formula (1.67 Å) and apply Bragg’s law for Ni (spacing $d = 0.91\,\text{Å}$):
    $n \lambda = 2d \sin\theta \; (n=1) \Rightarrow \lambda \approx 1.65 \,\text{Å}$.
  • Agreement confirms wave nature of electrons → first direct proof of matter waves.
• Additional evidences:
  • Electron diffraction from polycrystalline foils (Thomson), electron double-slit, neutron & X-ray diffraction parallels.

Bohr’s Quantisation via De Broglie

• Electron orbit circumference must contain integral number of matter wavelengths:
  $2\pi r = n\lambda = n \frac{h}{mv}$ ⇒ $mvr = \frac{nh}{2\pi}$.
• Provides natural derivation of Bohr’s postulated angular-momentum quantisation.
• Reinforces complementary wave/particle aspects (Bohr’s Complementarity Principle).

Heisenberg Uncertainty Principle (Consequence of Wave Packet)

• Single monochromatic matter wave ⇒ precise $p$ ( $p= \hbar k$ ) but delocalised $x$.
• Superposition to localise $x$ introduces spread in $k$ and hence $p$.
• Quantitative limit:
  $\Delta x\, \Delta p \ge \frac{h}{4\pi}$.
• Other conjugate pairs:
  • Energy–time: $\Delta E\, \Delta t \ge \frac{h}{4\pi}$.
  • Angular-momentum $L<em>z$ – azimuthal angle $\phi$: $\Delta L</em>z\, \Delta \phi \ge \frac{h}{4\pi}$.
• Demonstrated by thought experiments (electron microscope, diffraction through slit, etc.).

Practical & Philosophical Implications

• Basis of Schrödinger wave mechanics; matter wave $\psi$ encodes probability amplitude.
• Quantum objects exhibit particle OR wave properties depending on measurement (cannot display both simultaneously).
• Explains stability of atoms, tunnelling, quantisation in solids, etc.
• Experimental technologies derived:
  • Electron & neutron diffraction for crystal structure determination.
  • Electron microscopy, LEED, TEM.
  • Wave-based interpretation essential for semiconductor devices.
• Sets limit on simultaneous knowledge; reshapes deterministic worldview → probabilistic, yet strictly governed by quantum laws.

Quick Reference Formulae

• $\boxed{\lambda = \dfrac{h}{p}}\, ,\; p = mv$.
• $\boxed{\lambda = \dfrac{h}{\sqrt{2mE_{\text{K}}}}}$ (non-relativistic).
• Electron: $\boxed{\lambda(\text{Å}) = \dfrac{12.26}{\sqrt{V(\text{volts})}}}$.
• Bragg’s law: $n\lambda = 2d\sin\theta$.
• Phase velocity: $V<em>p = \dfrac{\omega}{k}$; Group velocity: $V</em>g = \dfrac{d\omega}{dk} = v$.
• Uncertainty relation: $\Delta x\Delta p \ge \tfrac{h}{4\pi}$.