Greens Functions, Propagators, and Time...
Transcript of Greens Functions, Propagators, and Time...
![Page 1: Greens Functions, Propagators, and Time Evolutionpages.physics.cornell.edu/~sethna/teaching/QuantumI/Lectures12/5_GreensPropagatorsTime...Greens Functions, Propagators, and Time Evolution](https://reader034.fdocuments.us/reader034/viewer/2022042106/5e8556a4b040220662121c61/html5/thumbnails/1.jpg)
Greens Functions, Propagators, and Time Evolution
Time evolution as operator:
Propagator for free particles is basis for Feynman diagrams
![Page 2: Greens Functions, Propagators, and Time Evolutionpages.physics.cornell.edu/~sethna/teaching/QuantumI/Lectures12/5_GreensPropagatorsTime...Greens Functions, Propagators, and Time Evolution](https://reader034.fdocuments.us/reader034/viewer/2022042106/5e8556a4b040220662121c61/html5/thumbnails/2.jpg)
Free particle propagator, one dimension
Simple Harmonic Oscillator
* Feynman's PhD thesis: from path integrals
![Page 3: Greens Functions, Propagators, and Time Evolutionpages.physics.cornell.edu/~sethna/teaching/QuantumI/Lectures12/5_GreensPropagatorsTime...Greens Functions, Propagators, and Time Evolution](https://reader034.fdocuments.us/reader034/viewer/2022042106/5e8556a4b040220662121c61/html5/thumbnails/3.jpg)
Gaussian Integrals and Cauchy's Theorem
Cauchy's Theorem: Integral around closed loop in complex plane zero if no singularities inside. (Exercise?) Close the contour. Real axis = diagonal plus arcs; check arcs go to zero at large |x|.
![Page 4: Greens Functions, Propagators, and Time Evolutionpages.physics.cornell.edu/~sethna/teaching/QuantumI/Lectures12/5_GreensPropagatorsTime...Greens Functions, Propagators, and Time Evolution](https://reader034.fdocuments.us/reader034/viewer/2022042106/5e8556a4b040220662121c61/html5/thumbnails/4.jpg)
More info on propagators:
* Like free propagator expand in 1 = sum |p><p|, expand in sum |E><E|:
* Advisor (PW Anderson) wrote: Green's function solves (E-H) G = delta(x'', x') = "1" => G = 1/(E-H)
making sense of his terse derivation!
![Page 5: Greens Functions, Propagators, and Time Evolutionpages.physics.cornell.edu/~sethna/teaching/QuantumI/Lectures12/5_GreensPropagatorsTime...Greens Functions, Propagators, and Time Evolution](https://reader034.fdocuments.us/reader034/viewer/2022042106/5e8556a4b040220662121c61/html5/thumbnails/5.jpg)
![Page 6: Greens Functions, Propagators, and Time Evolutionpages.physics.cornell.edu/~sethna/teaching/QuantumI/Lectures12/5_GreensPropagatorsTime...Greens Functions, Propagators, and Time Evolution](https://reader034.fdocuments.us/reader034/viewer/2022042106/5e8556a4b040220662121c61/html5/thumbnails/6.jpg)
Brief peek at Feynman diagrams & Dyson eqn