De Broglie's Hypothesis (AQA A Level Physics): Revision Note

De Broglie's Hypothesis of Wave-Particle Duality

What was DeBroglie's hypothesis?

• Louis DeBroglie hypothesised that all particles can behave both like waves and like particles, following Einstein's work with photons

• By equating two equations from Einstein, he derived an equation for the momentum of a photon:

$E=m{c}^2$ (more on this in Special Relativity)

$E=hf$ (the energy of a photon)

$m{c}^2=hf$

$mc=h\frac{f}{c}=\frac{h}{\lambda }$

• Where h  is Planck's constant, c  is the speed of light, m  is mass, λ  is wavelength and f  is frequency

• mc  is the momentum, p, of a photon - DeBroglie extended this idea to particles with mass to obtain the relation you should recall from Particles & Radiation:

$p=\frac{h}{\lambda }$

Finding the Wavelength of Accelerated Particles

• This idea can be applied to accelerated electrons to find their wavelength

  • Finding their momentum directly is difficult, but recall from The Discovery of the Electron that the work done on an electron by an electric field (eV) is equal to its kinetic energy - this can be used to find the electron's speed:

$eV=\frac{1}{2}m{v}^2$

$v=\sqrt{\frac{2eV}{m}}$

• This can be substituted into the momentum term in DeBroglie's hypothesis to then find wavelength:

$p=mv=m\sqrt{\frac{2eV}{m}}=\sqrt{m^2}\sqrt{\frac{2eV}{m}}=\sqrt{\frac{m^{2}2eV}{m}}=\sqrt{2meV}$

$\lambda =\frac{h}{\sqrt{2meV}}$

• The wavelength of the electron depends on the work done on it by the electric field, eV

  • From this equation, as eV  increases, λ  decreases

  • When the electron is accelerated to a higher speed, its DeBroglie wavelength decreases