What is the ratio of the kinetic energy of the photon to that of the electron
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h = Planck's constant, p = momentum of electron, m = mass of electron
v = velocity of electron, ν = frequency of wave of photon
λ = wavelength of photon E1 = kinetic energy of electron
E2 = Kinetic energy of photon c = speed of light
DeBroglie wavelength of electron = λ = h / p
p = m v => p² = 2 m E1 => p = √(2 m E1) => λ = h /√(2 m E1)
E1 = h² / (2 λ² m)
Photon
E2 = h ν = h c /λ => λ = h c /E2
=> E2 / E1 = h c / λ * 2 λ² m / h²
= 2 m c λ / h
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for example
energy of photon to energy of electron with the same wavelength 500 nm
= 2 * 9.11 * 10⁻³¹ * 3 * 10⁸ * 500 * 10⁻⁹ / 6.636 * 10⁻³⁴
= 0.008237
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wavelength of electron : λ = h / p = h / m v, v = velocity of electron
substitute this value in the ratio of energies.
E2 / E1 = 2 m c λ / h = 2 m c (h / m v) / h
=> E2 / E1 = 2 c / v
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