Question

What are de Broglie waves? Show that the wavelength of the electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).

Hint:

For a photon, the mass is zero, so we cannot use \(p = mv\). We must always use the relativistic relation \(p = E/c\).

Answer 1

de Broglie Waves
According to Louis de Broglie, every moving particle is associated with a wave known as a matter wave or de Broglie wave.
He proposed that not only light but also material particles such as electrons exhibit wave-like properties.
The de Broglie wavelength of a particle is given by:
λ = h / p
where h is Planck’s constant and p is the momentum of the particle.

Relation Between Electromagnetic Wavelength and de Broglie Wavelength of Photon
For electromagnetic radiation (light), energy of a photon is given by Planck’s equation:
E = hν
where ν is the frequency.
Also, from wave theory:
ν = c / λ
where c is the speed of light and λ is the wavelength of radiation.
Substituting ν:
E = h (c / λ)
Now, from Einstein’s mass–energy relation for photon:
E = pc
where p is the momentum of the photon.
Equating the two expressions for energy:
pc = h (c / λ)
Cancel c from both sides:
p = h / λ
Rearranging:
λ = h / p

Conclusion:
The wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).
Thus, photons also obey the de Broglie relation, confirming the wave–particle duality of light.