Energie

$E^{2}=m^{2}c^{4}+{\underline {p}}^{2}c^{2}$

mit Impuls

$\left|{\underline {p}}\right|=\gamma {{m}_{0}}\left|{\underline {v}}\right|$ folgt $E=\gamma {{m}_{0}}{{c}^{2}}$

Taylor-Entwicklung für kleine Geschwindigkeiten:

$Assumptions = c > 0 && Subscript[m, 0] >= 0 && p >= 0 && v >= 0;

Energy      := Sqrt[Subscript[m, 0]^2*c^4 + p^2*c^2];
EnergyAppox := Series[Energy, {p, 0, 5}] // Simplify;

Print["E" == EnergyAppox]

$E=c^{2}m_{0}+{\frac {p^{2}}{2m_{0}}}-{\frac {p^{4}}{8\left(c^{2}m_{0}^{3}\right)}}+O\left(p^{6}\right)$

De Brogli:

$E=\hbar \omega$