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The Poisson bracket {7|,|P|}
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In free space, U (r, t) must satisfy tne wave equation, VU - (1/)U/at = 0. Use the definition (12.1-21) to show that the mutual coherence function G(r1,r2, 7) satisfies a pair of partial differential cquations known as the Wolf equations, 1 PG = 0 vG – (12.1-24a) vG - 1 G = 0, (12.1-24b) where V and V are the Laplacian operators with respect to r, and r2, respectively. G(rı, r2, 7) = (U*(r1,t) U(r2, t + T)). (12.1-21) Mutual Coherence Function
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Consider a collection of N non-interacting one-dimensional harmonic oscillators, with total Hamiltonian H(p, q) = E"+mw²q}]: Li-1 [2m (a) Calculate the classical partition function, taking the phase-space element to be dpdq/t, where t is an arbitrary scale factor. (b) Obtain the entropy, internal energy, and heat capacity.
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Consider the matrix representation of Lx, Ly and L₂ for the case l = 1 (see Matrix Representation of Operators class notes pp. 11-12). (a) Construct the matrix representation of L² for l = 1. (b) What are the eigenvalues and corresponding eigenvectors of L²? (c) Are the eigenvectors of L² the same as those of L₂? Explain. (d) Compute L² |x; +1), where |x;+1) is the eigenvector of La corresponding to eigenvalue +ħ.