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de Broglie Wavelength

What you are seeing: the de Broglie wavelength λ=h/p\lambda = h/p plotted on a log-log axis against kinetic energy for five particle species: massless photon, electron, proton, neutron, and a 12^{12}C atom. The relativistic momentum pc=(T+mc2)2−(mc2)2pc = \sqrt{(T + mc^2)^2 - (mc^2)^2} is used so the curves are valid in both the non-relativistic (T≪mc2T \ll mc^2) and relativistic regimes.

Look for the slope-1/2 line at low energies (non-relativistic λ∝T−1/2\lambda \propto T^{-1/2}) breaking to a slope-1 line at high energies (ultra-relativistic λ∝T−1\lambda \propto T^{-1}). The transition happens at T∼mc2T \sim mc^2, which is why electrons go relativistic around 1 MeV but protons only around 1 GeV. The photon line is the universal relativistic asymptote, λ=hc/E\lambda = hc/E, a clean slope-1 line. Reference horizontal lines mark a typical atomic spacing (0.1 nm) and a nuclear scale (1 fm = 10−6^{-6} nm).

Figure 1. de Broglie wavelength versus kinetic energy across particle species. Method: closed-form relativistic λ=hc/pc\lambda = hc/pc.
species electron
log10(T/eV)2.00

WHAT TO TRY

  • Switch species and slide the energy: lambda = h/p drops as a particle speeds up, and heavier particles sit lower. A thermal neutron lands near atomic spacings, which is why it diffracts off crystals.
  • Push the energy until T/mc^2 approaches one: the curve bends as the relativistic momentum takes over from the classical sqrt(2mT).
  • Compare the electron and the carbon atom at the same energy: matter waves shrink as mass grows, the reason everyday objects never show interference.