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Unformatted text preview: December 15, 2006 Friday, 14:00 Hrs Quantum Physics 357: Final Examination Keshav Dasgupta 1 , Sangyong Jeon 2 Rutherford Physics Building, McGill University, Montreal, QC H3A 2T8, Canada Notes This question paper has eight pages including the cover page. The paper is divided into two parts: Part A and Part B . Part A has seven long answer questions, whereas Part B has ten relatively short answer questions. In Part A you should attempt three out of the seven questions and in Part B you should attempt five out of the ten questions. Please read carefully all the questions before you start writing the solutions. Make sure you write clearly and legibly. Mention all the necessary steps required to reach a particular solution. This is a closed book examination but calculators, regular dictionary and translation dictionary are allowed. You should also write your name and ID number on this and return it with your answer sheets. Time allowed three hours . Total credit: 100 marks . Name of the student: Student ID number: 1 Examiner 2 Associate Examiner Part A Note: Answer any three out of the following seven questions. Each question carry an equal credit of 15 marks . You are advised not to spend more than 25 mins on each questions. 1. Show that the energy of a massive particle in a 1+1 dimensional compact spacetime with a cylindrical topology is always quantised. How does the result change if the spacetime has the following topology: x t which basically means that the space is like an interval, with noncompact time. Can you compare this to any of the bound state problems that we studied? 6 + 6 + 3 = 15 2. A particle of mass m is trapped in a onedimensional potential of the form L V(x) x such that V = 0 for 0 x L and infinite elsewhere. Find the matrix representation of the Hamiltonian for this case. Find also the minimum energy of the particle in this potential. 12 + 3 = 15 3. A ndimensional Hilbert space is parametrised by basis vectors  + i i , i = 0 , ..., n that are eigenvectors of an operator A with eigenvalues + i . I define another operator B in this space that satisfies the following commutation relation with the operator A : [ B , A ] = B (1) 1 where is a given integer with 0 n . A third operator Q could be used to define a Hamiltonian H in the space as: H B = i Q (2) where the only thing known about Q is that it is hermitian. Find the matrix representationis that it is hermitian....
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 Spring '05
 KeshavDasgupta
 mechanics, Quantum Physics

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