CIE A Level Physics (9702) 2019-2021

Revision Notes

27.2.2 The de Broglie Wavelength

Calculating de Broglie Wavelength

  • Using ideas based upon the quantum theory and Einstein’s theory of relativity, de Broglie suggested that the momentum (p) of a particle and its associated wavelength (λ) are related by the equation:

Calculating de Broglie Wavelength equation 1

  • Since momentum p = mv, the de Broglie wavelength can be related to the speed of a moving particle (v) by the equation:

Calculating de Broglie Wavelength equation 2

  • Since kinetic energy E = ½ mv2
  • Momentum and kinetic energy can be related by:

Calculating de Broglie Wavelength equation 3

  • Combining this with the de Broglie equation gives a form which relates the de Broglie wavelength of a particle to its kinetic energy:

Calculating de Broglie Wavelength equation 4

  • Where:
    • λ = the de Broglie wavelength (m)
    • h = Planck’s constant (J s)
    • p = momentum of the particle (kg m s-1)
    • E = kinetic energy of the particle (J)
    • m = mass of the particle (kg)
    • v = speed of the particle (m s-1)

Worked example: de Broglie wavelength

de_Broglie_Wavelength_Question, downloadable AS & A Level Physics revision notes

Step 1:
Consider how the proton and electron can be related via their masses
The proton and electron are accelerated through the same p.d., therefore, they both have the same kinetic energy

Step 2:
Write the equation which relates the de Broglie wavelength of a particle to its kinetic energy:

Calculating de Broglie Wavelength Worked Example equation 1

Calculating de Broglie Wavelength Worked Example equation 2

Step 3:
Calculate the ratio:

Calculating de Broglie Wavelen

Calculating de Broglie Wavelength Worked Example equation 4

This means the de Broglie wavelength of the proton is 0.023 times smaller than that of the electron OR the de Broglie wavelength of the electron is about 40 times larger than that of the proton

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