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etotheipi

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Classically, the total energy of a particle/system ##E = T+V##. This is the usage seen in the Schrödinger equation, since E is an eigenvalue of ##\hat{H}## and ##\hat{H} = \hat{T} + \hat{V}##.

Then in special relativity, the total energy ##E## of a particle/system becomes the rest energy plus the kinetic energy, i.e. ##E = \sqrt{(mc^{2})^{2} + (pc)^{2}} = \gamma mc^{2} = (mc^{2}) + (\frac{1}{2}mv^{2} + \mathcal{O}(v^{4}))##. I understand that this is valid for free particles (i.e. if the potential is constant, set it arbitrarily to zero for the particle). I would have thought then that the full relation would be ##E = \sqrt{(mc^{2})^{2} + (pc)^{2}} + V##, but apparently this is incorrect because it is not a covariant equation?

And secondly, there are some formulae like the Planck-Einstein ##E=hf## relation (which, for example's sake might apply to an electron), where I can't decide whether ##E=T+V## or ##E=E_{0} + T## applies.

I wondered whether someone could help to clarify this? Thank you.

Then in special relativity, the total energy ##E## of a particle/system becomes the rest energy plus the kinetic energy, i.e. ##E = \sqrt{(mc^{2})^{2} + (pc)^{2}} = \gamma mc^{2} = (mc^{2}) + (\frac{1}{2}mv^{2} + \mathcal{O}(v^{4}))##. I understand that this is valid for free particles (i.e. if the potential is constant, set it arbitrarily to zero for the particle). I would have thought then that the full relation would be ##E = \sqrt{(mc^{2})^{2} + (pc)^{2}} + V##, but apparently this is incorrect because it is not a covariant equation?

And secondly, there are some formulae like the Planck-Einstein ##E=hf## relation (which, for example's sake might apply to an electron), where I can't decide whether ##E=T+V## or ##E=E_{0} + T## applies.

I wondered whether someone could help to clarify this? Thank you.

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