How to evaluate a finite product in path integrals of harmonic oscillator?
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The path integral in quantum mechanics may be defined as,
∫∞−∞…∫∞−∞exp{iℏΔt∑iL(xi,xi+1−xiΔt,i)}dx0…dxN∫−∞∞…∫−∞∞exp{iℏΔt∑iL(xi,xi+1−xiΔt,i)}dx0…dxN
where as the OP has noted, one 'slices' time into N+1N+1 segments and the idea is that the propagator is given by the formal limit as N→∞N→∞. Based on this paper, it seems that convergence has been established by Fujikawa in the norm operator topology, in B(L2(Rd))B(L2(Rd))providing the potential is smooth with at most quadratic growth (e.g. a harmonic oscillator).
This has been extended to show convergence remains, providing second space derivatives exist in Hd+1(Rd)Hd+1(Rd). These results show we can expect to indeed recover the original propagator in the continuum limit.
However, for any finite NN, we cannot expect to do anything but approximate the propagator; we can of course carry out the integration finitely many times simply. In fact, this is what is originally done to notice the pattern that emerges, which enables taking
∫∞−∞…∫∞−∞exp{iℏΔt∑iL(xi,xi+1−xiΔt,i)}dx0…dxN∫−∞∞…∫−∞∞exp{iℏΔt∑iL(xi,xi+1−xiΔt,i)}dx0…dxN
where as the OP has noted, one 'slices' time into N+1N+1 segments and the idea is that the propagator is given by the formal limit as N→∞N→∞. Based on this paper, it seems that convergence has been established by Fujikawa in the norm operator topology, in B(L2(Rd))B(L2(Rd))providing the potential is smooth with at most quadratic growth (e.g. a harmonic oscillator).
This has been extended to show convergence remains, providing second space derivatives exist in Hd+1(Rd)Hd+1(Rd). These results show we can expect to indeed recover the original propagator in the continuum limit.
However, for any finite NN, we cannot expect to do anything but approximate the propagator; we can of course carry out the integration finitely many times simply. In fact, this is what is originally done to notice the pattern that emerges, which enables taking
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It is easy to evaluate the green's function using path integral approach by evaluating classical action and using functional calculus method. Is it possible .
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