3.9 Wave-particle energy transfer 113portion, the forces are the same, but the distances are half as much, so the time to traverse
the potential well portion is
twell=2 v 0
a(
−1 +
√
1 +δ)
. (3.214)
Similarly, the time required to traverse the potential hill portion will be
thill=2 v 0
a(
1 −
√
1 −δ)
(3.215)
so the average velocity for particlesBandDwill be
vavgB,D=ad/v 0
√
1 +δ−√
1 −δ. (3.216)
These particles moveslowerthan the injection velocity, but the effect is second order inδ.
The average velocity of the four particles will be
vavg =1
4
(
vAavg+vBavg+vavgC +vDavg)
(3.217)
=
ad
4 v 0[
1
−1 +
√
1 + 2δ+
1
1 −
√
1 − 2 δ+
2
√
1 +δ−√
1 −δ]
.
Ifδis small this expression can be approximated as
vavg ≃ad
4 δv 0[
1
1 −δ/2 +δ^2 / 2+
1
1 +δ/2 +δ^2 / 2+
2
1 +δ^2 / 8]
(3.218)
=
ad
2 δv 0[
1 +δ^2 / 2
1 + 3δ^2 / 4+
1
1 +δ^2 / 8]
≃ v 0[
1 − 3 δ^2 / 16]
so that the average velocity of the four representative particles issmallerthan the injection
velocity. This effect is second order inδand shows that a group of particles injected
at random locations with identical velocities into a sawtooth periodic potential will, on
average, be slowed down.
3.9.3 Slowing down, energy conservation, and average velocity
The sawtooth potential analysis above shows that is necessary to be very careful about what
is meant by energy and average velocity. Each particle individually conserves energy and
regains its injection velocity when it returns to the phase at which it was injected. However,
the average velocities of the particles are not the same as the injection velocities. ParticleA
has an average velocity higher than its injection velocity whereas particlesB,CandDhave
average velocities smaller than their respective injection velocities. The average velocity of
all the particles is less than the injection velocity so that the average kinetic energy of the
particles is reduced relative to the injection kinetic energy. Thus the average velocity of a
group of particles slows down in a periodic potential, yet paradoxically individual particles
do not lose energy. The energy that appears to be missing is contained in the instantaneous
potential energy of the individual particles.