a) de Broglie wavelength of a 4-MeV proton: 14.3 fm
b) de Broglie wavelength of a 40-GeV electron: 0.031 fm
Explanation:
a)
The de Broglie wavelength of an object is given by
(1)
where
h is the Planck constant
p is the momentum of the particle
Here we want to find the de Broglie wavelength of a 4-MeV proton. The rest of mass of the proton in MeV is
And since , this means that the proton is non-relativistic. So its kinetic energy is related to its momentum by
which means
where
is the kinetic energy
is the proton mass
Substituting, we find
b)
In this case, the electron has kinetic energy of 40 GeV, while the rest mass of an electron is
Since , the electron is ultra-relativistic: so we can rewrite its energy as
The equation (1) can also be rewritten as
where c is the speed of light. The quantity at the denominator is the energy, so
where:
is the energy of the electron
And substituting, we find:
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