• 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:

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

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

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

• 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

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:

Step 3:
Calculate the ratio:

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