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Unformatted text preview: q cl ( t ) = 0 when j ( t ) = 0. When solving classical EOM you may ±nd Fourierintegral decomposition q ( t ) = ∞ i∞ dE 2 π ei ¯ h E t q E useful. b. (10 pts) Use the result of part a to show that the twopoint function for the harmonic oscillator without the external force is given by a  T ˆ q ( t 1 ) ˆ q ( t 2 )  A = i ¯ h 2 m ∞ i∞ dE 2 π ei ¯ h E ( t 1t 2 ) E 2 − ¯ h 2 ω 2 + i ǫ . (1) 1 c. (10 pts) Rederive the twopoint function in Eq. (1) by using creation and annihilation operators. For the simple harmonic oscillator (without the external force) write ˆ q ( t ) = r ¯ h 2 m ω b ˆ a ei ω t + ˆ a † e i ω t B and use commutation relations for ˆ a and ˆ a † ([ˆ a, ˆ a † ] = 1, all other commutators are zero) to obtain Eq. (1). 2...
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 Fall '10
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 Work, Quantum Field Theory, Trigraph, external force

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