hw2 - F ∂ ∂p G-∂ ∂p F ∂ ∂q G ³´ FG where the...

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HOMEWORK # 2 Physics 6572 Friday, 9/19/08; due 9/26/08 1. Problem 11, p. 110, Chapter 2, Gottfried and Yan. You should study Section 2.5(f) on gauge invariance before working on this problem. 2. Consider the operators U ( p 0 ,q 0 ) = e i/ ¯ h ( pq 0 - qp 0 ) , where p 0 and q 0 are c-numbers, and p and q are the canonical variables which satisfy the commutation relation [ q,p ] = i ¯ h . a) Show that the trace of the product U ( p 0 ,q 0 ) U ( p 00 ,q 00 ) is tr { U ( p 0 ,q 0 ) U ( p 00 ,q 00 ) } = 2 π ¯ ( p 0 - p 00 ) δ ( q 0 - q 00 ) . b) An arbitrary operator F ( p,q ) is expanded in terms of a linear combination of U ( p 0 ,q 0 ) as follows F ( p,q ) = Z d p 0 d q 0 2 π ¯ h f ( p 0 ,q 0 ) U ( p 0 ,q 0 ) , where f ( p 0 ,q 0 ) is a c-number coefficient. Find an expression for the coefficient f ( p 0 ,q 0 ) in terms of F ( p,q ) and U ( p 0 ,q 0 ). c) Find the trace of the operator F ( p,q ) in terms of f ( p 0 ,q 0 ). 3. Consider two arbitrary operator functions F ( q,p ) and G ( q,p ). Show that GF = exp ± - i ¯ h ² ∂q
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Unformatted text preview: F ∂ ∂p G-∂ ∂p F ∂ ∂q G ³´ FG, where the subscript F (or G ) attached to a partial derivative indicates that the particular derivative applies to operator F (or G ). In the classical limit ¯ h → 0, we need to keep only the first two terms in the exponential series, and GF = FG-i ¯ h ² ∂F ∂q ∂G ∂p-∂F ∂p ∂G ∂q ³ , i.e. 1 i ¯ h [ F,G ] = [ F,G ] P.B. , where [ F,G ] P.B. denotes the Poisson Bracket in classical mechanics. The operator derivatives are defined as ∂F ( q,p ) ∂q = lim Δ q → 1 Δ q { F ( q + Δ q,p )-F ( p,q ) } , where Δ q is an infinitesimal c-number....
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