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a) At nite temperature, the total tunneling probability can be written as
X
1
�
W = e; En Ane; h SW KB (En)
(2)
n with = 1=T . Assuming that the levels En are closely spaced, so that the sum over n
can be replaced by an integral, and ignoring the energy dependence of the prefactor An,
show that the maximum in the sum corresponds to the energy E for which
KB
� ; @ SW@E (E ) =
(3) Note that the quantity has a meaning of the classical travel time. Thus at a nite
temperature the bounce paths that give the leading contribution to the escape rate are
periodic functions of the imaginary time, x(t + ) = x(t).
b) Discuss the temperature dependence of the escape rate for the cubic parabola (1).
Start at high temperature (small ), and show that in this case the saddle point trajectory
is constant, x(t) = xmax , where xmax is the barrier top coor...
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 Fall '04
 LeonidLevitov
 Physics

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