Why are de-Broglie waves associated with a moving football not visible?
The wavelength, X, of a photon and the de-Broglie wavelength of an electron have the same value.
Show that the energy of the photon is $\frac{2 λmc}{h}$ times the kinetic energy of the electron, where m, c and h have their usual meanings for electron.

Solution

Because of large mass of a football, the wavelength associated with a moving football is small. So its wave nature is not visible.

De-Broglie wavelength of electron,

λ = \frac{h}{p}

Momentum of electron,

p = \frac{h}{λ}

Kinetic energy of electron is,

= \frac{1}{2} {mv}^{2} = \frac{p^{2}}{2 m} = \frac{h^{2}}{2 mλ} [∵ p = mv]

Energy of a photon

= \frac{hc}{λ}

∴

\frac{Kinetic energy of photon}{Kinetic energy of electron} = \frac{hc}{λ} . \frac{2 {mλ}^{2}}{h^{2}} = \frac{2 λmc}{h}

∴

Energy of photon = \frac{2 λmc}{λ} \times K . E . of electron

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