**Problem 1.15 Suppose you wanted to describe an unstable particle, that spon- taneously disintegrates with a "lifetime" t. In that case the total probability of finding the particle somewhere should not be constant, but should decrease at (say) an exponential rate: too P(t) = | V(x, 1)1²dx = e~1/*. A crude way of achieving this result is as follows. In Equation 1.24 we tacitly assumed that V (the potential energy) is real. That is certainly reasonable, but it leads to the "conservation of probability" enshrined in Equation 1.27. What if we assign to V an imaginary part: V = Vo – ir, where Vo is the true potential energy and r is a positive real constant? (a) Show that (in place of Equation 1.27) we now get d P 2Г = -- P dt (b) Solve for P(t), and find the lifetime of the particle in terms of r.

**Problem 1.15 Suppose you wanted to describe an unstable particle, that spon- taneously disintegrates with a "lifetime" t. In that case the total probability of finding the particle somewhere should not be constant, but should decrease at (say) an exponential rate: too P(t) = | V(x, 1)1²dx = e~1/*. A crude way of achieving this result is as follows. In Equation 1.24 we tacitly assumed that V (the potential energy) is real. That is certainly reasonable, but it leads to the "conservation of probability" enshrined in Equation 1.27. What if we assign to V an imaginary part: V = Vo – ir, where Vo is the true potential energy and r is a positive real constant? (a) Show that (in place of Equation 1.27) we now get d P 2Г = -- P dt (b) Solve for P(t), and find the lifetime of the particle in terms of r.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

**Problem 1.15 Suppose you wanted to describe an unstable particle, that spon-
taneously disintegrates with a "lifetime" t. In that case the total probability of
finding the particle somewhere should not be constant, but should decrease at
(say) an exponential rate:
too
P(t) =
| V(x,1)1²dx = e=1/*.
-0-
A crude way of achieving this result is as follows. In Equation 1.24 we tacitly
assumed that V (the potential energy) is real. That is certainly reasonable, but it
leads to the "conservation of probability" enshrined in Equation 1.27. What if we
assign to V an imaginary part:
V = Vo – ir,
where Vo is the true potential energy and r is a positive real constant?
(a) Show that (in place of Equation 1.27) we now get
dP
21
= --P.
dt
(b) Solve for P(1), and find the lifetime of the particle in terms of r.

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