851HW9_09 - α = 0 the coherent state | α =0 A is exactly...

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PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 9 Topics Covered: parity operator, coherent states, tensor product spaces. Some Key Concepts: unitary transformations, even/odd functions, creation/annihilation operators, displaced vacuum states, displacement operator, tensor-product states, tensor-product operators, Schmidt decomposition, configuration space. 1. The Parity Operator: [20 pts] Determine the matrix element ( x | Π | x ) and use it to simplify the identity Π = integraltext dxdx | x )( x | Π | x )( x | , then use this identity to compute Π 2 , Π 3 , and Π n . From these results find an expression for S ( u ) = exp[Π u ] cosh u in the form f ( u ) + g ( u )Π. What is ( x | S ( u ) | ψ ) ? Express your answer in terms of ψ even ( x ) = 1 2 ( ψ ( x ) + ψ ( x )) and ψ odd ( x ) = 1 2 ( ψ ( x ) ψ ( x )). Compute ( x | S (0) | ψ ) , lim u →∞ ( x | S ( u ) | ψ ) , and lim u →−∞ ( x | S ( u ) | ψ ) . 2. [15 pts]The coherent state | α ) is defined by | α ) = e −| α | 2 / 2 n =0 α n n ! | n ) , where the states {| n )} are the harmonic oscillator energy eigenstates. First, show that for
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Unformatted text preview: α = 0, the coherent state | α =0 A is exactly equal to the harmonic oscillator ground-state, | n =0 A . Then show that any other coherent state can be created by acting on the ground-state, | A , with the ‘displacement operator’ D ( α ), i.e. show that | α A = D ( α ) | A , where D ( α ) := e αA † − α * A (1) You may need the Zassenhaus formula e B + C = e B e C e − [ B,C ] / 2 , which is valid only when [ B, [ B,C ]] = [ C, [ B,C ]] = 0. What is D ( α 2 ) | α 1 A ? 3. [15 pts] Consider a system described by the Hamiltonian H = p κ ( A + A † ). Use your results from the previous problem to determine | ψ ( t ) A for a system initially in the ground-state, | ψ (0) A = | A . 4. [10pts each] Cohen Tannoudji, pp341-350: problems 3.6, 3.7, 3.11 1...
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