# single paeticle gree

Jun 21st, 2015
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By the Heisenberg equation of motion, we can find time-evolution of the operators,

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Equation Chapter 1 Section 1Preliminaries: you have,11\* MERGEFORMAT (.)Green function equivalent to wavefunction. Retarded and advanced Green functions are defined as,12\* MERGEFORMAT (.)Sidenote: Some textbooks (see the Appendix's reference to SSP 10, (1.27)) write (1.2) as,13\* MERGEFORMAT (.)For non-interacting particles: (1.2) is equivalent to the single-particle wavefunction, which is,14\* MERGEFORMAT (.)2nd type of single-particle Green-function: the "greater" and "lesser" Green's functions,15\* MERGEFORMAT (.)Looking at (1.2) and (1.5), we see that and span the same solution space. Therefore: there is a linear transformation between them, and it is,16\* MERGEFORMAT (.)The Green functions (1.6) can propagate a single particle, , using the many particle H.Basis switch: you go from a realspace basis to an arbitrary1 one by using,17\* MERGEFORMAT (.)Example: consider a translation-invariant system. For starters, consider the retarded Green function of (1.2), and to study translational-invariance consider the specific functional form . Using (1.7), go to a momentum (k) space representation,18\* MERGEFORMAT (.)Example: compute the of (1.5) for free electrons. Then, it's natural to use (1.7) to go to k-space. In doing so, we notice that G> uses rather than a commutator (in contrast to GR).The k-basis: The Hamiltonian is diagonal, . Thus the Green function , and so, the Green function appears in the k-basis

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