PHYS4221(Fall2010)_ProblemSet_1

PHYS4221(Fall2010)_ProblemSet_1 - muting operators and ξ...

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PHYS4221 Quantum Mechanics I Fall term of 2010 Problem Set No.1 (Due on September 20, 2010) 1. Consider a charge q of mass m moving in an electromagnetic field. The Lagrangian of such a system is given by L = 1 2 m~v · ~v - ( ~ r,t ) + q c ~v · ~ A ( ~ r,t ) where c is the speed of light, and φ ( ~ r,t ) and ~ A ( ~ r,t ) are the scalar and vector potentials of the electromagnetic field respectively. Show that this Lagrangian gives the correct equation of motion of the particle: d dt ( m~v ) = q ~ E + ~v c × ~ B ! . Then obtain the corresponding Hamiltonian of the system and the Hamilton’s equations of motion. 2. A particle of mass m is in the environment of a force field with components: F z = - Kz and F y = F x = 0 for some constant K . Write down the Hamiltonian of the particle in Cartesian coordinates. What are the constants of motion? Use the fact that the Hamiltonian itself is also constant to obtain the orbit. What is the Hamiltonian in cylindrical coordinates? What are the constants of motion? Prove the following theorems: 3. Theorem 1 (Baker-Hausdorff formula) : If A and B are two fixed noncom-
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Unformatted text preview: muting operators and ξ is a parameter, then exp (-ξA ) B exp ( ξA ) = B + ξ 1! [ B,A ] + ξ 2 2! [[ B,A ] ,A ] + ξ 3 3! [[[ B,A ] ,A ] ,A ] + ······ . [ Hint : Expand exp (-ξA ) B exp ( ξA ) in a power series of ξ .] 4. Theorem 2 : If A and B are two noncommuting operators and ξ is a parameter, then exp (-ξA ) B n exp ( ξA ) = [exp (-ξA ) B exp ( ξA )] n for some integer n , and exp (-ξA ) F ( B ) exp ( ξA ) = F (exp (-ξA ) B exp ( ξA )) where F ( B ) is an arbitrary function of B . 1 5. Theorem 3 : If the operators a and b satisfy the commutation relation [ a,b ] = 1, then show that exp (-ξba ) a exp ( ξba ) = a exp ( ξ ) exp (-ξba ) b exp ( ξba ) = b exp (-ξ ) exp ±-1 2 ξ ² b 2-a 2 ³ ´ a exp ± 1 2 ξ ² b 2-a 2 ³ ´ = a cosh ( ξ ) + b sinh ( ξ ) exp ±-1 2 ξ ² b 2-a 2 ³ ´ b exp ± 1 2 ξ ² b 2-a 2 ³ ´ = b cosh ( ξ ) + a sinh ( ξ ) . —— End —— 2...
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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PHYS4221(Fall2010)_ProblemSet_1 - muting operators and ξ...

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