show that the wavelength of electromagnetic radiation is equal to the de-Broglie wavelength of its quantum(photon).

To demonstrate that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum, which we refer to as a photon, we need to delve into some fundamental concepts of physics, particularly quantum mechanics and wave-particle duality.

Understanding Electromagnetic Radiation

Electromagnetic radiation, such as light, can be described as waves that travel through space. The wavelength (λ) of this radiation is a key characteristic, representing the distance between successive peaks of the wave. The relationship between the speed of light (c), frequency (ν), and wavelength is given by the equation:

c = νλ

Here, c is approximately 3 x 10⁸ m/s in a vacuum. This equation shows that the wavelength is inversely proportional to the frequency; as frequency increases, wavelength decreases, and vice versa.

Introducing the De Broglie Wavelength

Now, let's shift our focus to the de Broglie wavelength, which applies to all matter, including photons. According to de Broglie's hypothesis, every particle can exhibit wave-like behavior, and its wavelength can be calculated using the formula:

λ = h/p

In this equation, h is Planck's constant (approximately 6.626 x 10^-34 Js), and p is the momentum of the particle. For a photon, the momentum can be expressed in terms of its energy (E) using the relation:

p = E/c

Connecting the Concepts

For a photon, the energy can also be related to its frequency by the equation:

E = hν

Substituting this expression for energy into the momentum equation gives:

p = hν/c

Calculating the De Broglie Wavelength for a Photon

Now, we can substitute this expression for momentum into the de Broglie wavelength formula:

λ = h/p = h/(hν/c) = c/ν

Notice that this result shows that the de Broglie wavelength of a photon is equal to the wavelength derived from the electromagnetic wave equation:

λ = c/ν

Conclusion: Equivalence of Wavelengths

Thus, we have shown that the wavelength of electromagnetic radiation (λ = c/ν) is indeed equal to the de Broglie wavelength of its quantum (photon). This equivalence beautifully illustrates the wave-particle duality of light, where photons exhibit both wave-like and particle-like properties. Understanding this relationship is crucial in fields such as quantum mechanics, optics, and even modern technologies like lasers and semiconductors.