Supplement 1.5: Mass, Energy and Momentum of Particles and Photons (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 = \sqrt{p^{2} c^{2} + m_{o}^{2} c^{4}}

Detailed calculus ↓

Squaring the equation for the relativistic particle energy:

E^{2} = m^{2} c^{4} = \frac{m_{o}^{2} c^{4}}{1 - \frac{v^{2}}{c^{2}}} = \frac{m_{o}^{2} c^{6}}{c^{2} - v^{2}}

Rearranging, $E^{2} c^{2} = E^{2} v^{2} + m_{o}^{2} c^{6}$

dividing by c² and replacing E² in the first term on the right side by m²c⁴ gives:

E^{2} = m^{2} c^{2} v^{2} + m_{o}^{2} c^{6}

Replacing in the first term on the right side the product m²v² by p² and taking the square root yields the given equation.

Energy and momentum of photons

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

E = p c

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

E = h f

and so it follows the momentum equation of photons:

p = \frac{h}{c} f = \frac{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 $ω = 2 π f$ , and the wavenumber k, with $k = 2 π / λ$ .

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

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

Particles having a non-zero rest mass

have an energy $E = m c^{2}$ and a momentum $p = m v$
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 $E = h f$ or $E = ℏ ω$
and a momentum $p = \frac{h}{λ}$ or $p = ℏ k$

Understanding Spectra from the Earth

Supplement 1.5: Mass, Energy and Momentum of Particles and Photons (2/2)

The particle energy ... cont. from previous page

Energy and momentum of photons