# 21.6 THE WAVE-PARTICLE DUALITY The concept of wave-particle duality describes how light and matter exhibit properties of both waves and particles. Light demonstrates wave properties like interference, diffraction, and polarization, but also particle properties as seen in the , where it behaves as discrete packets of energy called photons. In 1924, Louis De Broglie extended this duality, proposing that particles of matter can also exhibit wave-like behavior. ## Wave-Particle Duality of Light Light behaves as a wave in phenomena like diffraction and polarization, but as a particle (photon) in phenomena like the photoelectric effect. This means light has a dual nature. ## De Broglie's Hypothesis Proposed by Louis De Broglie in 1924, this hypothesis states that all matter has wave-like properties. Just as photons have momentum, moving particles have an associated wavelength. ## De Broglie Wavelength The wavelength ($\lambda$) associated with a moving particle is given by the De Broglie relation. This wavelength is inversely proportional to the particle's momentum ($p$). **Formula**: $$ \lambda = \frac{h}{p} $$ where $p = mv$, so the formula can also be written as: $$ \lambda = \frac{h}{mv} $$ | Symbol | Name | Value / Unit | | :--- | :--- | :--- | | $\lambda$ | De Broglie Wavelength | meters (m) | | $h$ | Planck's Constant | $6.63 \times 10^{-34} \text{ J s}$ | | $p$ | Momentum | kg m/s | | $m$ | Mass | kilograms (kg) | | $v$ | Velocity | m/s | ## Observability of Wavelength - **Macroscopic Objects**: Objects with large mass (e.g., a 0.5 kg ball) have extremely small De Broglie wavelengths (e.g., $1.3 \times 10^{-34}$ m), which are too small to be measured or observed. - **Microscopic Particles**: Particles like electrons have small masses, resulting in wavelengths (e.g., $7.3 \times 10^{-10}$ m) that are comparable to the spacing between atoms in a crystal. This allows their wave nature to be observed through effects like diffraction and interference. ## Practical Application The wave nature of electrons is the theoretical basis for the **electron microscope**, which allows for high-resolution imaging of microscopic structures like viruses and bacteria. ## Worked Example **Q:** The speed of a rifle bullet is $1500 \mathrm{~m} \mathrm{~s}^{-1}$. If its mass is 20 g, then find the De Broglie wavelength associated with it. Is this wavelength observable? Give reason. **A:** - **Given**: $m = 20 \text{ g} = 0.02 \text{ kg}$, $v = 1500 \text{ m/s}$ - **Formula**: $\lambda = \frac{h}{mv}$ - **Calculation**: $$ \lambda = \frac{6.63 \times 10^{-34}}{(0.02)(1500)} = \frac{6.63 \times 10^{-34}}{30} = 2.21 \times 10^{-35} \text{ m} $$ - **Conclusion**: This wavelength is **not observable**. It is incredibly small, far smaller than the size of an atomic nucleus, making it impossible to detect with any current technology.