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WOLFRAM|DEMONSTRATIONS PROJECT

Classical Barrier Crossing and Phase Space

initial conditions
x
0
-2.5
p
0
0.5
time
0.001
This Demonstration illustrates the classical motion of a particle governed by the Hamiltonian
H=
2
p
2m
+
-α
2
x
e
. Here
x(t)
and
p(t)
are obtained by solving Hamilton's equations of motion, subject to the initial conditions
x
0
=x(0)
and
p
0
=p(0)
. The three panels animate synchronously: (1) the motion of the particle as it approaches the potential, (2) the phase space trajectory, and (3) the time series of
x(t)
and
p(t)
. In the upper two panels, the points with minimum momentum (turning points) are labeled with blue
x
's.
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