If photon is an elementary particle, how can different photons have different energy, if $E=mc^2$ and all photons have (or don't have) the same mass and the speed of photon is constant shouldn't it mean all photons have the same amount of energy?
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$E = mc^2$ is only true for particles at rest (momentum $p = 0$). The full formula is $$E = \sqrt{m^2c^4 + p^2c^2}$$.
In these formulas, $m$ is the rest mass (or just "mass"). Photons don't have mass, so for them the formula becomes $E = pc = \frac{hc}{\lambda} = h\nu$ where $\nu$ is the frequency.
Therefore, the energy of a photon is proportional to its frequency.
Tobias Brandt
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I know that everyone recognizes $E = mc^2$ as the emblem of relativity, but the expressions is either incomplete or only applies to massive particles (depending on how you understand the $m$ that appears there).
The full and complete expression is $$ (mc^2)^2 = E^2 - (pc)^2 \,,$$ where $m$ should be understood to be the invariant rest mass.
This allows you to recover either interpretation of the shorter form for massive particles and $E = pc$ for massless particles.
