CIE A Level Physics复习笔记22.2.2 The de Broglie Wavelength

The de Broglie Wavelength

  • De Broglie proposed that electrons travel through space as a wave
    • This would explain why they can exhibit behaviour such as diffraction
  • He therefore suggested that electrons must also hold wave properties, such as wavelength
    • This became known as the de Broglie wavelength
  • However, he realised all particles can show wave-like properties, not just electrons
  • So, the de Broglie wavelength can be defined as:

            The wavelength associated with a moving particle

  • The majority of the time, and for everyday objects travelling at normal speeds, the de Broglie wavelength is far too small for any quantum effects to be observed
  • A typical electron in a metal has a de Broglie wavelength of about 10 nm
  • Therefore, quantum mechanical effects will only be observable when the width of the sample is around that value
  • The electron diffraction tube can be used to investigate how the wavelength of electrons depends on their speed
    • The smaller the radius of the rings, the smaller the de Broglie wavelength of the electrons
  • As the voltage is increased:
    • The energy of the electrons increases
    • The radius of the diffraction pattern decreases
  • This shows as the speed of the electrons increases, the de Broglie wavelength of the electrons decreases

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:


  • 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

A proton and an electron are each accelerated from rest through the same potential difference.

  • Mass of a proton = 1.67 × 10–27 kg
  • Mass of an electron = 9.11 × 10–31 kg