The relationship between the energy of a particle and its wavelength arises from the wave-particle duality of quantum mechanics. According to the de Broglie hypothesis, proposed by Louis de Broglie in 1924, particles such as electrons or photons exhibit wave-like properties and have associated wavelengths.
The de Broglie wavelength (λ) of a particle is given by the equation:
λ = h / p
where λ is the wavelength, h is Planck's constant (a fundamental constant in quantum mechanics), and p is the momentum of the particle.
The energy (E) of a particle is related to its momentum (p) through the equation:
E = p^2 / (2m)
where E is the energy and m is the mass of the particle.
By substituting the expression for momentum (p) in terms of wavelength (λ) into the equation for energy (E), we get:
E = h^2 / (2mλ^2)
From this equation, we can see that the energy of a particle is inversely proportional to the square of its wavelength. This implies that particles with shorter wavelengths (higher frequency) have higher energies, while particles with longer wavelengths (lower frequency) have lower energies.
The reason particles have wavelengths is a fundamental aspect of quantum mechanics. It arises from the wave-particle duality, which suggests that all objects, including particles, can exhibit both particle-like and wave-like properties. The wavelength associated with a particle reflects its wave-like nature and is a fundamental property of quantum mechanics.
It's important to note that the wave-like behavior of particles becomes more noticeable at the microscopic scale, such as electrons and photons. For macroscopic objects, the wavelength associated with their motion is incredibly small and not typically observable.