Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).
According to the de Broglie relation, the wavelength associated with a particle of momentum is given by . This relation is satisfied by photons as well.
For a photon, the momentum is given by , where is Planck's constant, is the frequency, and is the speed of light. Substituting this into the de Broglie relation, we get:
Since the wavelength of electromagnetic radiation is defined as , the de Broglie wavelength of the photon is equal to the wavelength of the electromagnetic radiation.
Explanation
The textbook context explicitly derives this equality in Eq. (11.7). It starts with the de Broglie hypothesis (Eq. 11.5) and applies the photon momentum formula (Eq. 11.6). By substituting the photon momentum into the de Broglie equation, the term cancels out, leaving , which is the standard definition of the wavelength for electromagnetic waves. This demonstrates the consistency of the wave-particle duality concept for radiation.
Solution Steps
Step 1: State the de Broglie relation for a photon: .
Step 2: State the momentum for a photon from context: .
Step 3: Substitute the momentum into the de Broglie relation: .
Step 4: Simplify the equation to show .
Step 5: Conclude that since is the wavelength of electromagnetic radiation, the two wavelengths are equal.