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# 582-2008-4 - Physics 582 Fall Semester 2008 Professor...

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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 one-dimensional 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 Euler-Lagrange 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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582-2008-4 - Physics 582 Fall Semester 2008 Professor...

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