This leads to the useful approximation:įor electron energies less than 20 keV and proton energies less than 38 MeV this expression has an error less than 1%. UsingĪn expression for pc in terms of the relativistic kinetic energy can be obtained: Application to pion decayįor most ordinary processes the kinetic energy is much less than the rest mass energy, so the first term can usually be neglected. The quantity pc is proportional to the relativistic momentum but has energy units, and it is useful in many high energy calculations. The relativistic energy expression for a particle is: Low velocity calculation of DeBroglie wavelength
For low speed calculations like the baseball example, it might be more accurate to use the non-relativistic calculation. Under those conditions, small differences between large numbers occur in the calculations.
The corresponding DeBroglie wavelength isĬaution! Since this calculation was designed for speeds which are a significant fraction of the speed of light, it may lose some accuracy for very low speeds. Where m e= electron rest mass and m p= proton rest mass,Ĭorresponding to velocity v = x10^ m/s = c, The following calculation uses the full relativistic expressions for kinetic energy, etc. (This is why the limiting resolution of an electron microscope is much higher than that of an optical microscope.) For massive particles with kinetic energy KE which is much less than their rest mass energies:įor an electron with KE = 1 eV and rest mass energy 0.511 MeV, the associated DeBroglie wavelength is 1.23 nm, about a thousand times smaller than a 1 eV photon. This is particularly appropriate for comparison with photon wavelengths since for the photon, pc=E and a 1 eV photon is seen immediately to have a wavelength of 1240 nm. DeBroglie Wavelength DeBroglie WavelengthĪ convenient form for the DeBroglie wavelength expression is
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