advertisement

HJi . Problem 2.1.6. ,Show that the propagator of a harmonic oscillator has the form \ G(xb,t,x.,0) = A(t) exp imwo 2 [(xb2 + xa) cos(wot) — 2xbx.]) 2 sin(two) Use the normalization condition or the path integral to show that A(t) = mwo 27ri sin(two) ) 2 1/ s 0 J3, A Problem 2.3.6. Spin waves: Consider a spin-S quantum spin chain H = Ei Jsisi+,For J < 0, the classical ground state is a ferromagnetic state with Si = Si. For J > 0, it is an anti-ferromagnetic state with Si = Si(—)i. 1. Write down the action for the spin chain. 2. Find the equation of motion for small fluctuations 6Si = Si — Si around the ferromagnetic ground state. Transform the equation of motion to frequency and • momentum space and find the dispersion relation. Show that w 6c k2 for small k. 52()i around the 3. Find the equation of motion for small fluctuations 6Si = Si— anti-ferromagnetic ground state. Find the dispersion relation. Show that w cc 1k — in. for k near to in. _ .— •