Wave Nature of Matter 🌊

Light shows both wave and particle behaviors—interference and diffraction need waves, while photoelectric and Compton effects need photons. This mix naturally leads to the question: do particles (electrons, protons, etc.) also act like waves? The answer—yes!—opens the door to matter waves. :contentReference[oaicite:0]{index=0}

1  de Broglie’s Bold Hypothesis 💡

In 1924 Louis de Broglie suggested that every moving particle carries a wave whose wavelength depends on its momentum. He wrote:

$$\lambda \;=\;\frac{h}{p}\;=\;\frac{h}{m\,v}\qquad(11.5)$$

Here \(h\) is Planck’s constant, \(p\) the particle’s momentum, \(m\) its mass, and \(v\) its speed. This is the de Broglie relation. The left-hand side (\(\lambda\)) belongs to waves, while the right-hand side (\(p\)) belongs to particles—showing duality in one elegant stroke. :contentReference[oaicite:1]{index=1}

2  Special Case: Photons 🔆

For a photon, the momentum is \(p = \dfrac{h\nu}{c}\) (11.6). Substituting in (11.5) gives

$$\lambda \;=\;\frac{h\,c}{p}\;=\;\frac{c}{\nu}\qquad(11.7)$$

So the de Broglie wavelength of a photon is exactly the ordinary wavelength of the light itself—nice symmetry! :contentReference[oaicite:2]{index=2}

3  What Changes \(\lambda\)? 🧐

  • Heavier particles (\(m\) large) ⟹ smaller \(\lambda\)
  • Faster particles (\(v\) large) ⟹ smaller \(\lambda\)

Example: A 0.12 kg ball moving at 20 m s-1 has \(p = 2.40\;\text{kg m s}^{-1}\) and \(\lambda = 2.76\times10^{-34}\,\text{m}\)—far too tiny to detect. :contentReference[oaicite:3]{index=3} That’s why your cricket ball never shows diffraction!

4  Quick Calculations 🔢

Electron vs Football:

  • Electron with \(v = 5.4\times10^{6}\,\text{m s}^{-1}\)
    ⟹ \(p = 4.92\times10^{-24}\,\text{kg m s}^{-1}\),  \(\lambda = 0.135\,\text{nm}\) (similar to X-ray spacing). :contentReference[oaicite:4]{index=4}
  • 150 g ball at 30 m s-1
    ⟹ \(p = 4.50\,\text{kg m s}^{-1}\),  \(\lambda = 1.47\times10^{-34}\,\text{m}\) (utterly negligible). :contentReference[oaicite:5]{index=5}

5  When Do Matter Waves Matter? 🔍

In everyday life, momenta are huge and de Broglie wavelengths are minuscule, so wave effects hide. In the atomic world—where \(m\) is tiny—\(\lambda\) reaches nanometers, matching inter-atomic distances and making diffraction experiments (like electron diffraction) possible. :contentReference[oaicite:6]{index=6}

High-Yield Ideas for NEET 🔥

  1. de Broglie relation: $$\lambda = \dfrac{h}{p} = \dfrac{h}{m v}$$
  2. For photons, \(\lambda\) from de Broglie equals the light’s own wavelength.
  3. \(\lambda\) decreases with increasing mass or speed—so wave behavior is prominent only for microscopic particles.
  4. Typical electron \(\lambda\) (≈ 0.1 nm) matches crystal lattice spacing—basis of electron diffraction.
  5. de Broglie’s idea links matter and radiation, paving the way for quantum mechanics.

😊 Happy studying—matter waves are real waves in the quantum playground!