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A powerful way to calculate expected values is to rely on symmetry. A) Show that if the energy E is symmetric about the point x = a [i.
1. A powerful way to calculate expected values is to rely on symmetry.
A) Show that if the energy E is symmetric about the point x = a [i.e., E(a−x) = E(a+x)], then ˂x> = a
B) Consider the small thermal fluctuations of an elastic fiber. The energy of the fiber is given by
E(x) = A(x−x0)2, where x is the length of the fiber and A is a constant related to the fiber stiffness. Without performing any integrals, show that average length of the fiber is ˂x> = x0. State any assumptions that you make.
C) Let's impose an electric field E0 to stretch the fiber. The energy of the fiber is now given by E(x) = A(x−x0)2 − BE0x, where B is a positive constant. Find ˂x>. (Can you do it without evaluating any integrals?)
