Starting from the equation E 2 =(mc 2 ) 2 + (pc) 2 , show that in theclassical l

Question

Starting from the equation E2 =(mc2)2 + (pc)2, show that in theclassical limit (pc << mc2) the energy approachesthe classical kinetic energy plus the rest mass energy. Hint: you might want to use the binomial theorem: (1 +x)n ~ 1 + xn, when x is small. Starting from the equation E2 =(mc2)2 + (pc)2, show that in theclassical limit (pc << mc2) the energy approachesthe classical kinetic energy plus the rest mass energy. Hint: you might want to use the binomial theorem: (1 +x)n ~ 1 + xn, when x is small.

Explanation / Answer

E2 = (mc2)2 + (pc)2= (mc2)2 [1 +(pc/(mc2))2] = (mc2)2[1 + (p/(mc))2] = (mc2)2 [1 +(mv/(mc))2] = (mc2)2 [1 +(v/c)2] E = mc2 [1 + (v/c)2]1/2 E ˜ mc2 [1 +v2/(2c2)] = mc2 + mv2/2 = the rest mass energy mc2 plus the classical kineticenergy mv2/2.

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