22.2.2 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

• 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, mv² can be re-written as p² / m (from p=mv)
• Therefore, 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)