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First teaching 2020

Last exams 2024

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The de Broglie Wavelength (CIE A Level Physics)

Revision Note

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Katie M

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Katie M

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What is 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:

Calculating de Broglie Wavelength equation 2

  • 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:

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

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

Determine the ratio: fraction numerator d e space B r o g l i e space w a v e l e n g t h space o f space t h e space p r o t o n over denominator d e space B r o g l i e space w a v e l e n g t h space o f space t h e space e l e c t r o n end fraction

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

2.5.4 De Broglie Wavelength Worked Example

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Katie M

Author: Katie M

Expertise: Physics

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.