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Mathematics for Physics

68 de Broglie Wavelength

The wavelength associated with a moving material particle, equal to Planck’s constant divided by the particle’s momentum, that is used to assign a wavelength to a particle from its momentum.

Note: Also called the de Broglie relation.

definition [d] (de Broglie Wavelength) From Knight: if a material particle of momentum \(p = mv\) has a wave-like nature, then its wavelength must be given by

  • \(\lambda = \dfrac{h}{p} = \dfrac{h}{mv}\) ,

where \(h\) is Planck’s constant. This is called the de Broglie wavelength.

where

  • \(\lambda\) is the de Broglie wavelength.
  • \(p\) is the momentum of the particle.
  • \(m\) is the mass of the particle.
  • \(v\) is the speed of the particle.
  • \(h\) is Planck’s constant.

68.1 Elementary Example

68.1.1 Simple

An electron with momentum \(p\) has de Broglie wavelength \(\lambda = h/p\).

\[ p = 1.0 \times 10^{-24}\,\mathrm{kg\cdot m/s} \]

\[ \lambda = \dfrac{6.626 \times 10^{-34}}{1.0 \times 10^{-24}} = 6.626 \times 10^{-10}\,\mathrm{m} \]

where

  • \(\lambda\) is the de Broglie wavelength.
  • \(p\) is the particle momentum.

68.1.2 General

For three momenta, wavelength falls as momentum rises.

\[ p_{1} = 1 \times 10^{-24},\quad p_{2} = 2 \times 10^{-24},\quad p_{3} = 3 \times 10^{-24}\,\mathrm{kg\cdot m/s} \]

\[ \lambda_{i} = \dfrac{h}{p_{i}} \]

where

  • each \(\lambda_{i}\) is the de Broglie wavelength for momentum \(p_{i}\).