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Unformatted text preview: HW#2 —Phys374—Spring 2008 www.physics.umd.edu/grt/taj/374c/ due before class, Friday, Feb. 15 Prof. Ted Jacobson (301)405-6020, [email protected] 1. In class we made a Taylor series expansion of f ( x ) = 1 / (1- x ) about x = 0, and ob- served using Mathematica that this expansion seemed to converge only in the interval | x | < 1. Nevertheless, we may expand around points outside this interval. Find the Taylor series about the point x = 3. Write a formula for the n th term in the series. (It turns out that this series converges in the interval | x- 3 | < 2.) [5 pts.] 2. Relativistic energy The relativistic relation between energy E , 3-momentum p , and rest mass m is E 2 = p 2 c 2 + m 2 c 4 , (1) where c is the speed of light. In class we found the first few terms in the expansion of the function E ( p ) in powers of p/mc , E = mc 2 + p 2 / 2 m- p 4 / 8 m 3 c 2 + ... . The first term is the rest energy, and the second has the form of nonrelativistic kinetic energy. Those terms would give a good approximation for low momentum, i.e. when p mc . In this problem look at the opposite limit. Find the approximate form of E ( p ) for very high momentum p mc (equivalently, E mc 2 ). More precisely, expand in powers of mc/p , keeping only the m-independent term (which governs massless particles, like photons) and the leading order mass dependence. (Relativistic momentum is mv/ p 1- v 2 /c 2 , so it can be much greater than mc when v is very close to c , and it can be nonzero in the limit m → 0 only if v → c .) [5 pts.] 3. Diatomic molecules The Lennard-Jones potential, a simple approximation for the interaction energy be-...
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