# HJi exp [(xb + xa) cos(wot) — 2xbx.]) ```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 &lt; 0,
the classical ground state is a ferromagnetic state with Si = Si. For J &gt; 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.
_ .—
•
```