Deriving the De Broglie Wavelength Equation

Matter exhibits both particle and wave nature. The derivation of the De Broglie equation establishes the fundamental relationship between these two natures of a particle.

Louis de Broglie's Hypothesis on Dual Nature

In 1924, the French physicist Louis de Broglie proposed that electrons also possess particle and wave characteristics, just as photons or light. According to his hypothesis, every particle exhibits dual characteristics. Furthermore, he indicated that the path of electrons is wavy, similar to light having a definite frequency. For this groundbreaking theory, De Broglie received the Nobel Prize in Physics in 1929.

The De Broglie Equation and Confirmation

The experiment of cathode-ray diffraction by George Paget Thomson and the Davisson–Germer experiment, which explicitly applied to electrons, confirmed De Broglie’s equation.

Thus, the derivation of the De Broglie equation addresses the wave properties of matter, mainly electrons.

λ = h/mv

Variables in the Equation:

• λ: Wavelength
• h: Planck’s constant
• m: Particle’s mass
• v: Velocity of the particle

Steps for Deriving the De Broglie Equation

For the derivation of De Broglie’s equation, we need to follow two fundamental equations (theories):

1. Einstein’s Equation of Mass and Energy

This equation relates energy (E) to mass (m):

E = mc²

Where:

• E = energy
• m = mass
• c = speed of light in vacuum

2. Planck’s Equation Indicating Energy from Waves

This equation relates the energy (E) of a quantum to its frequency (ν):

E = hν

Where:

• E = energy
• h = Planck’s constant (value: 6.62607 x 10^-34 J·s)
• ν = frequency

Equating Energy Expressions

De Broglie considered the above two energies equal, based on his belief that particles and waves display similar traits (wave-particle duality). Based on this hypothesis:

mc² = hν

We know that frequency (ν) is related to the speed of light (c) and wavelength (λ) by the relation ν = c/λ.

Substituting this into the equation:

mc² = h x c/λ

Rearranging to solve for wavelength (λ):

λ = h/mc

This equation is known as the De Broglie equation for a photon. It is also known as the matter-wave equation.

Generalizing for Particle Velocity

The actual particles (like electrons) do not travel at the speed of light (c). Hence, De Broglie substituted the particle's velocity (v) for the speed of light (c) to generalize the equation:

λ = h/mv

Alternative Form using Momentum

Since the product of mass (m) and velocity (v) is defined as momentum (p), we can also write the De Broglie Equation as:

λ = h/p

(Note: For a photon, p = mc is the momentum.)