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Unformatted text preview: Physics 216 Midterm Exam Spring 2010 MIDTERM EXAM INSTRUCTIONS : This is an open book exam. You are permitted to consult the textbooks of Shankar and Baym, your handwritten notes, and any class handouts that are posted to the course website. One mathematical reference book is also permitted. No other consultations or collaborations are permitted during the exam. In order to earn total credit for a problem solution, you must show all work involved in obtaining the solution. However, you are not required to re-derive any formulae that you cite from the textbook or the class handouts. The point value of each problem is indicated in the square brackets below. 1.  Suppose we define G ( t ) integraldisplay dxK ( x,t, ; x, 0) (1) where K ( x,t ; x ,t ) is the propagator. Assume that the system has a time-independent Hamiltonian and a discrete energy level spectrum. (a) Prove that the Fourier transform of G , tildewide G ( E ) lim i planckover2pi1 integraldisplay G ( t ) e iEt/ planckover2pi1 e t dt (2) has poles at all the discrete energy levels of the system. Take to be a positive infinitesimal quantity. HINT: For a time-independent Hamiltonian, the time-evolution operator has a simple form. Writing K ( x,t ; x ,t ) as a coordinate-space matrix element of the time evolution operator, insert a complete set of energy eigenstates. Then, computeoperator, insert a complete set of energy eigenstates....
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- Spring '09