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De Broglie's equation

After the discovery of the photoelectric effect it was realised that waves can possess particle-like properties, and a search was made to see if the reverse was true, could particles behave like waves.

In 1923 Louis de Broglie proposed that a particle of mass m travelling with a velocity v would have a wavelength λ given by the equation:

Wavelength of a particle:    λ = h/mv

where h is the Planck constant. The intensity of the wave represented the probability of the particle being and at that point.
The formula allows us to calculate the wavelength associated with a moving cricket ball.

Example problems
1. Find the wavelength of an electron moving at 3x107 ms-1. (mass of electron = 9 x 10-31 kg).
λ= h/p = [6.63x10-34] / [9.00x10-31 x 3.00x107] = 6.63x10-34 / 2.70x10-23 = 2.46x10-11 m = 0.025 nm

2. Find the wavelength of a cricket ball of mass 0.15 kg moving at 30 m s-1.
λ = [6.7x 10-34]/0.15x30 = 1.49 x 10-34 m - a very small number!

This is not really a very sensible answer or indeed a very sensible problem because of course a cricket ball is composed of billions upon billions of individual particles each having their own discrete wavelength.
However for individual electrons the formula has a real meaning; and since an electron accelerated through a potential difference of V volts will gain electrical energy (E = eV) and hence kinetic energy mv2 we can calculate the wavelength of the electron.

Therefore, substituting values for the charge and mass for the electron:

Electron wavelength (λ) = [12.27x10-10]/V1/2

Example problems
Calculate the wavelength of an electron accelerated through a potential difference of (a) 10 kV and (b) 100 V.

(a) λ = 1.23x10-11 m
(b) λ = 1.23x10-10 m.

This is the true beginning of wave-particle duality, since sometimes the particle would behave like a wave and sometimes like a particle. Even more strangely, the way in which it behaved seemed to be influenced by the nature of the experiment used.

We can also use the formula to calculate the 'mass' of a quantum of light, or photon. For yellow light of wavelength 600 nm the calculated mass is 3.7 x 10-36 kg. It is interesting to compare this with a mass of 9x10-31 kg for the electron. Using these ideas it is possible to calculate the recoil velocity of an atom that emits a quantum of light.


© Keith Gibbs