The de Broglie hypothesis, proposed by Louis de Broglie in 1924, states that every moving particle of matter is associated with a wave, known as a matter wave. This revolutionary idea established that matter exhibits both wave-like and particle-like properties, a concept known as wave–particle duality. Matter waves, also called de Broglie waves, provide a theoretical explanation for several quantum phenomena and form a fundamental principle of modern quantum mechanics.
The de Broglie wavelength (𝜆) of a particle is given by the equation:\[ \lambda =\frac{h}{p} \]
From Einstein’s Mass-Energy Relation
\[ E = mc^2 \hspace{5cm} (1)\]
\[ E = \frac{hc}{\lambda} \hspace{5cm} (2) \]
\[ mc^2 = \frac{hc}{\lambda} \]
Solving for \( \lambda \):
\[ \lambda = \frac{h}{mc} \]
If the particle is moving with velocity \( v \), replacing \( c \) with \( v \):
\[ \lambda = \frac{h}{mv} \]
In terms of momentum \( p \):
\[ \lambda = \frac{h}{p} \]
Kinetic Energy of the Particle
\[ KE = \frac{1}{2} mv^2 \hspace{5cm} (3)\]
Multiplying by \( m \) on both sides:
\[ 2mKE = m^2 v^2 \hspace{5cm} (4)\]
De Broglie Wavelength
\[ \lambda = \frac{h}{mv} \]
Squaring both sides:
\[ \lambda^2 = \frac{h^2}{m^2 v^2} \]
Substituting Eqn 4 in the above Eqn:
\[ \lambda^2 = \frac{h^2}{2mKE} \]
Solving for \( \lambda \):
\[ \lambda = \frac{h}{\sqrt{2mKE}} \]
If a charged particle 'e' is accelerated through a potential difference 'V', the kinetic energy is given by
\[ KE = eV \hspace{5cm}(5)\]
kinetic energy of the particle:
\[ KE = \frac{1}{2} mv^2 \hspace{5cm} (6)\]
Equating equations 5 and 6,
\[ eV = \frac{1}{2} mv^2 \]
multiplying m on both sides:
\[ 2m eV = m^2v^2 \hspace{5cm} (7) \]
The debroglie wavelength is given by:
\[ \lambda = \frac{h}{mv} \]
Squaring both sides
\[ \lambda^2 = \frac{h^2}{m^2v^2} \]
Substituting Eqn 7 in above Eqn :
\[ \lambda^2 = \frac{h^2}{2meV} \]
\[ \lambda = \frac{h}{\sqrt{2meV}} \hspace{5cm} (7) \]
The final equation we get after substituting the all constant values (h, m and e) in Eqn 7,
\[ \lambda = \frac{12.26}{\sqrt{V}} Å \]
The De-Broglie Hypothesis introduced the revolutionary idea of wave-particle duality, stating that every moving particle has an associated wavelength called the De-Broglie wavelength. This wavelength is inversely proportional to momentum, meaning lighter and faster particles have shorter wavelengths. The hypothesis successfully explained why electrons in atoms show diffraction patterns like waves while behaving as particles in experiments. It laid the foundation for quantum mechanics and was later experimentally verified by the Davisson–Germer experiment.
c) Momentum
b) h/√(2meV)
b) Davisson–Germer experiment
c) Very short
a) Only light particles