الملحق رقم 1.5: الكتلة و الطاقة و الزخم للجسيمات و الفوتونات (2\2)

The particle energy      ... cont. from previous page

Squaring the relations of energy and momentum and combining both equations yields an equation for the relativistic energy of a particle in the following form:

E= p 2 c 2 + m o 2 c 4
Detailed calculus

Energy and momentum of photons

The rest energy of photons is zero. With mo=0 it follows from the equation of the relativistic particle energy given above:

E=pc

On the other hand, according to Planck and Einstein the energy of photons is:

E=hf ,

and so it follows the momentum equation of photons:

p= h c f= h λ


In Supplement 1.2 on page 2 we introduced two quantities, which can be used instead of the frequency f and wavelength λ: the circular frequency ω, with ω=2πf , and the wavenumber k, with k= 2π /λ .

Instead of the Planck constant h one often uses the so-called reduced Planck constant =h/ 2π (pronounced “h-bar”).

With these parameters we can write: E=ω , p=k .

Particles having a non-zero rest mass

  • have an energy E=m c 2 and a momentum p=mv
  • have a relativistic mass, which increases with velocity and becomes infinity at the speed of light; therefore they cannot propagate with the speed of light

Photons

  • have a rest mass which is zero and propagate always with the speed of light
  • have an energy        E=hf or  E=ω
    and a momentum     p= h λ  or  p=k