The de Broglie Wavelength (Cambridge (CIE) A Level Physics)

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

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:

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

• Using kinetic energy , momentum and kinetic energy can be related by:

Energy:  

Momentum:

• 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)