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412 chapter 150.22Probability0.150.1quantumclassicalPosition100.05-10-40 -20 0 20 40Position-20 20 40 60 80 100TimeFigure 15.8 Left: The probability distribution for the harmonic oscillator ground state asdetermined with a path-integral calculation (the classical result has maxima at the twoturning points). Right: A space-time trajectory used as a quantum path.15.8.1 Bound-State Wave Function (Theory)Although you may be thinking that you have already seen enough expressions forthe Green’s function, there is yet another one we need for our computation. Let usassume that the Hamiltonian ˜H supports a spectrum of eigenfunctions,˜Hψ n = E n ψ n ,each labeled by the index n. Because ˜H is Hermitian, the solutions form a completeorthonormal set in which we may expand a general solution:ψ(x, t)=∞∑c n e −iEnt ψ n (x), c n =n=0∫ +∞−∞dx ψ ∗ n(x)ψ(x, t =0), (15.46)where the value for the expansion coefficients c n follows from the orthonormalityof the ψ n ’s. If we substitute this c n back into the wave function expansion (15.46),we obtain the identityψ(x, t)=∫ +∞−∞∑dx 0 ψn(x ∗ 0 )ψ n (x)e −iEnt ψ(x 0 ,t=0). (15.47)Comparison with (15.39) yields the eigenfunction expansion for G:nG(x, t; x 0 ,t 0 =0)= ∑ nψ ∗ n(x 0 )ψ n (x)e −iEnt . (15.48)We relate this to the bound-state wave function (recall that our problem is tocalculate that) by (1) requiring all paths to start and end at the space positionx 0 = x, (2) by taking t 0 =0, and (3) by making an analytic continuation of (15.48)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 412