# Two different formulas for the calculation of energy in QM

The De-Broglie approach tells us that the momentum of a wave is $$p=\frac{h}{\lambda}.$$ Thus for an electromagnetic wave ($$m=0$$, phase velocity $$c$$) the Energy is: $$E=pc=h\frac{c}{\lambda}=h\nu.$$ For a particle with mass $$m$$, which can also be described as a wave with wavelength $$\lambda$$ (e.g. electron) the kinetic Energy is calculated with: $$E_{\rm kin}=\frac{1}{2}mv^2=\frac{p^2}{2m}=\frac{h^2}{2m\lambda^2}.$$

• The first one $$E=\frac{hc}{\lambda}$$ should be used for massless ($$m=0$$) particle.
• The second one $$E=\frac{h²}{2m\lambda²}$$ should be used for massive ($$m \neq 0$$) particles.

The first equation gives the energy of a photon (zero mass) in terms of its wavelength, $$\lambda$$.

The second gives the kinetic energy of a particle of mass m (moving at a speed much less than $$c$$, the speed of light) in terms of its de Broglie wavelength, $$\lambda$$.