First slide

Matter waves (DC Broglie waves)

Question

The energy of a photon is equal to the kinetic energy of a proton. The energy of the photon is E. Let $λ_{1}$ , be the de-Broglie wavelength of the proton and $λ_{2}$ , be the wavelength of the photon. The ratio $λ_{1} / λ_{2}$ is proportional to

Moderate

A

$E^{0}$
B

$E^{1 / 2}$
C

$E^{- 1}$
D

$E^{- 2}$

Solution

We know that $λ = \frac{h}{mv} = \frac{h}{p}$

$Here \frac{1}{2} {mv}^{2} = \frac{p^{2}}{2 m} = E or p = \sqrt{2 mE} λ_{1} = \frac{h}{\sqrt{2 mE}} and λ_{2} = \frac{hc}{E} therefore \frac{λ_{1}}{λ_{2}} = \frac{h}{\sqrt{2 mE}} \times \frac{E}{hc} = \frac{E^{1 / 2}}{\sqrt{2 mc}} or \frac{λ_{1}}{λ_{2}} \propto E^{1 / 2}$

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