Physics 582, Fall Semester 2008
Professor Eduardo Fradkin
Problem Set No. 4:
Path Integrals in Quantum mechanics and in Quan
tum Field Theory
Due Date: October 26, 2008; 9:00 pm
1
Path Integral for a particle in a double well
potential.
Consider a particle with coordiante
q
, mass
m
moving in the onedimensional
double well potential
V
(
q
)
V
(
q
) =
λ
(
q
2
−
q
2
0
)
2
(1)
In this problem you will use the path integral methods,in imaginary time, that
were discussed in class to calculate the matrix element,
(
q
0
,
T
2
 −
q
0
,
−
T
2
)
=
(
q
0

e
−
1
¯
h
HT
 −
q
0
)
(2)
to
leading order
in the semiclassical expansion, in the limit
T
→ ∞
.
1. Write down the expression of the imaginary time path integral that is
appropriate for this problem. Write an explict expression for the Euclidean
Lagrangian (
i.e.,
the Lagrangian in imaginary time). How does it differ
from the Lgrangian in real time? Make sure that you specify the initial
and final conditions. Do not calculate anything yet!
2. Derive the EulerLagrange equation for this problem (always in imaginary
time).
Compare it with the equation of motion in real time.
Find the
explicit solution for the trajectory (in imaginary time) that satisfies the
initial and final conditions. Is the solution unique? Explain. What is the
physical interpretation of this trajectory and of the amplitude?
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 Fall '08
 Leigh
 mechanics, Vector Space, Quantum Field Theory, General Relativity, real time, Spacetime, Fundamental physics concepts

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