Consider a particle with mass m moving on a one-dimensional periodic wire (a circle). The Hamil-
tonian is given by H = 5% + V(:1:) as usual, but we restrict to a: E [-L, L) and identify a: = -L to
a: = L. We consider a potential that depends on two parameters, A > 0 and n, where
K, :1: + L if :1: < 0
V(:r) = A332 + ( )’ . + Constant (1)
K.(L-:1:), 1f:z: >0
a) 3pts Depending on A and Is, this system either has one ground state or two degenerate ground
states. For which values of parameters are there two degenerate ground states? At what values of
a: are these ground states located?
b) 2pts Assume the parameters admit two degenerate ground states. There are two possible ways
to tunnel from a given ground state to the other: what are they? Depending on /\ and Ii, tunneling
via one path may dominate over the other. Using the symmetries of the potential, determine the
parameter ranges in /\ and h: when each tunneling path dominates.
c) 5pts Compute the associated Euclidean actions SE(xd (7)) corresponding to each possible tun-
neling process. Show that your answer is consistent with (b).
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