B.2 Double-Well Potential
409
Fig. B.3 Quartic double-well
system: x versus
V (x) = −2Bx 2 + x 4
Fig. B.4 Phase space plot for
the quartic double-well
system
H o =
p 2
2m
− 2Bx
2
+ x
4
= E o .
(B.17)
The double-well potential, V (x) = −2Bx 2 + x 4 , is plotted in Fig. B.3 and phase
space orbits are shown in Fig. B.4. Trajectories with energy E o < 0 will be trapped
in one of the two wells and cannot travel across the barrier, while trajectories with
energy E o > 0 can travel freely across the barrier. This is the analog of libration
and rotation, respectively, in the pendulum. We shall call trajectories trapped or
untrapped according to whether or not they can cross the barrier. It is necessary to
consider the two cases separately.
B.2.1 Below the Barrier—(E o < 0)
The momentum for the case E o < 0 can be written in the form
p = ±
2m(E o + 2Bx 2 − x 4 ) = ±
2m(f 2 − x 2 )(x 2 − e 2 ),
(B.18)
where