〈 Berger | Dillon 〉

Haha nice, but don't you need a (1/2) with the pc?

Don't think so. E is the time component and pc the space part of the momentum four-vector.

When you Taylor-expand E = sqrt(m^2 c^4 + p^2 c^2) you get E = mc^2 + p^2 / 2m + ... Maybe this 1/2 is what @mathematicsprof was talking about? It shows up automatically - no need to stick it in by hand.

Hmm what do the rest of the terms of the Taylor expansion look like? Any large contribution to total energy that would be often ignored?

The other contributions get large for objects moving near the speed of light, so you can't ignore them in particle accelerators.

Ah right (found when googled). What about fine structure of atoms? Some electrons in atoms travel in relativistic speeds.

that's what John is saying .. if they are traveling at relativistic speeds you can't ignore the higher order terms

Yeah, in hydrogen the electron moves about 1/137 the speed of light, but for a nucleus with N protons the innermost electrons move at about N/137 times the speed of light - and for some other reason we don't expect atoms with roughly N > 137 to exist.