Q.Derive de-Broglie's wave equation for dual nature of matter. Give its significance.

The Intuition: Why Would an Electron Have a Wavelength?

Before 1924, physics had a clean split: light was a wave (interference, diffraction), and electrons were particles (they had mass, charge, followed Newton's laws). Then came experiments that blurred the line. Light, which everyone knew was a wave, also behaved like a stream of particles (photons) in the photoelectric effect. Einstein had shown that a photon's energy is $E = hf$, and its momentum is $p = h/\lambda$.

Louis de Broglie asked a bold, symmetrical question: if light — a wave — can behave like a particle, why can't a particle — like an electron — behave like a wave? Nature, he argued, might be symmetric. Every moving object should have a wavelength associated with it.

The Precise Statement

For any particle with momentum $p = mv$, the associated de Broglie wavelength $\lambda$ is:

where:

• $h = 6.63 \times 10^{-34} \ \text{J·s}$ (Planck's constant)
• $m$ is the mass of the particle (in kg)
• $v$ is its speed (in m/s)

For an electron, this is not a mathematical trick — it is a real, measurable wave property. The electron's wave nature is not visible in everyday life because the wavelength is incredibly small for macroscopic objects, but for an electron moving at typical speeds, it becomes comparable to atomic spacings in crystals.

Why It Matters for an Electron

Take an electron accelerated through a potential difference $V$. Its kinetic energy comes from the electrical work done:

Solving for velocity and plugging into the de Broglie relation gives a very handy formula:

For $V = 100$ volts, this works out to about $1.23 \times 10^{-10}$ m — roughly the spacing between atoms in a crystal. That is why electrons can be diffracted by crystals, exactly like X-rays. The wave nature of the electron is not a philosophical curiosity; it is the operating principle of the electron microscope.

The Key Insight …