Phy5670/RPA: Difference between revisions
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<math> G_{ph} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') = \lim_{t_{\beta} \rightarrow t^{+}} | <math> G_{ph} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') = \lim_{t_{\beta} \rightarrow t^{+}} | ||
\lim_{t_{\gamma} \rightarrow t'^{+}} G_{II} (\alpha t, \delta t', \beta t_{\beta}, \gamma t_{\gamma}) | \lim_{t_{\gamma} \rightarrow t'^{+}} G_{II} (\alpha t, \bar{\delta} t', \bar{\beta} t_{\beta}, \gamma t_{\gamma}) | ||
= -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [a_{\beta}^{H+}(t) a_{\alpha}^{H}(t) a_{\gamma}^{H+} (t') a_{\delta}^{H} (t')] | \psi_{o}^{N} \rangle </math> (Eq. 1) | = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [a_{\bar{\beta}}^{H+}(t) a_{\alpha}^{H}(t) a_{\gamma}^{H+} (t') a_{\bar{\delta}}^{H} (t')] | \psi_{o}^{N} \rangle </math> (Eq. 1) | ||
where "ph" means "particle-hole pairs". Substituting the explicit form of the Heisenberg operators and inserting a complete set of N-particle state one has | where "ph" means "particle-hole pairs". Substituting the explicit form of the Heisenberg operators and inserting a complete set of N-particle state one has | ||
<math> G_{ph} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\beta}^{+}e^{-iHt/\hbar} e^{iHt/\hbar}a_{\alpha}e^{-iHt/\hbar} e^{iHt'/\hbar}a_{\gamma}^{+}e^{-iHt'/\hbar} e^{iHt'/\hbar}a_{\delta}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle </math> | <math> G_{ph} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}}^{+}e^{-iHt/\hbar} e^{iHt/\hbar}a_{\alpha}e^{-iHt/\hbar} e^{iHt'/\hbar}a_{\gamma}^{+}e^{-iHt'/\hbar} e^{iHt'/\hbar}a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle </math> | ||
<math> = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\beta} a_{\alpha}e^{-iHt/\hbar} e^{iHt'/\hbar}a_{\gamma}^{+} a_{\delta}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle </math> | <math> = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}} a_{\alpha}e^{-iHt/\hbar} e^{iHt'/\hbar}a_{\gamma}^{+} a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle </math> | ||
<math> = -\frac{i}{\hbar} \sum_{n}^{} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\beta} a_{\alpha}e^{-iHt/\hbar} | \psi_{n}^{N}\rangle \langle \psi_{n}^{N} | e^{iHt'/\hbar}a_{\gamma}^{+} a_{\delta}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle </math> | <math> = -\frac{i}{\hbar} \sum_{n}^{} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}} a_{\alpha}e^{-iHt/\hbar} | \psi_{n}^{N}\rangle \langle \psi_{n}^{N} | e^{iHt'/\hbar}a_{\gamma}^{+} a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle </math> | ||
<math> = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\beta} a_{\alpha}e^{-iHt/\hbar} | \psi_{o}^{N}\rangle \langle \psi_{o}^{N} | e^{iHt'/\hbar}a_{\gamma}^{+} a_{\delta}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle | <math> = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}} a_{\alpha}e^{-iHt/\hbar} | \psi_{o}^{N}\rangle \langle \psi_{o}^{N} | e^{iHt'/\hbar}a_{\gamma}^{+} a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle | ||
-\frac{i}{\hbar} \sum_{n \neq 0}^{} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\beta} a_{\alpha}e^{-iHt/\hbar} | \psi_{n}^{N}\rangle \langle \psi_{n}^{N} | e^{iHt'/\hbar}a_{\gamma}^{+} a_{\delta}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle</math> | -\frac{i}{\hbar} \sum_{n \neq 0}^{} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}} a_{\alpha}e^{-iHt/\hbar} | \psi_{n}^{N}\rangle \langle \psi_{n}^{N} | e^{iHt'/\hbar}a_{\gamma}^{+} a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle</math> | ||
<math> = -\frac{i}{\hbar} \langle \psi_{o}^{N} | a_{\beta}^{+} a_{\alpha} | \psi_{o}^{N} \rangle | <math> = -\frac{i}{\hbar} \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{o}^{N} \rangle | ||
\langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\delta} | \psi_{o}^{N} \rangle | \langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle | ||
- \frac{i}{\hbar} [ \sum_{n \neq 0}^{} \theta (t-t') e^{i(E_{o}^{N} - E_{n}^{N})(t-t')/\hbar} \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle | |||
+ \sum_{n \neq 0}^{} \theta (t'-t) e^{i(E_{o}^{N} - E_{n}^{N})(t'-t)/\hbar} \langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{o}^{N} \rangle ] </math> (Eq. 2) | |||
+ \sum_{n \neq 0}^{} \theta (t'-t) e^{i(E_{o}^{N} - E_{n}^{N})(t'-t)/\hbar} \langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\delta} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\beta}^{+} a_{\alpha} | \psi_{o}^{N} \rangle ] </math> (Eq. 2) | |||
where the definition of the time-ordering operator in terms of step functions is used also. The so-called '''polarization propagator''' is defined by Eq. (2) which includes the excited states only: | where the definition of the time-ordering operator in terms of step functions is used also. The so-called '''polarization propagator''' is defined by Eq. (2) which includes the excited states only: | ||
Revision as of 16:35, 4 December 2010
Polarization Propagator
To study excited states in meny-fermion systems, the limit of the two-particle (tp) propagator is used
(Eq. 1)
![{\displaystyle G_{ph}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')=\lim _{t_{\beta }\rightarrow t^{+}}\lim _{t_{\gamma }\rightarrow t'^{+}}G_{II}(\alpha t,{\bar {\delta }}t',{\bar {\beta }}t_{\beta },\gamma t_{\gamma })=-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[a_{\bar {\beta }}^{H+}(t)a_{\alpha }^{H}(t)a_{\gamma }^{H+}(t')a_{\bar {\delta }}^{H}(t')]|\psi _{o}^{N}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c2830e55f9700e8c6a5f48d10ae0d139f50252f)
where "ph" means "particle-hole pairs". Substituting the explicit form of the Heisenberg operators and inserting a complete set of N-particle state one has
![{\displaystyle G_{ph}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')=-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}^{+}e^{-iHt/\hbar }e^{iHt/\hbar }a_{\alpha }e^{-iHt/\hbar }e^{iHt'/\hbar }a_{\gamma }^{+}e^{-iHt'/\hbar }e^{iHt'/\hbar }a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f08f5565f31607ed3226daba86d4b7e09dc4c634)
![{\displaystyle =-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}a_{\alpha }e^{-iHt/\hbar }e^{iHt'/\hbar }a_{\gamma }^{+}a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a41f196ebb5be486dc64261885743597829846da)
![{\displaystyle =-{\frac {i}{\hbar }}\sum _{n}^{}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}a_{\alpha }e^{-iHt/\hbar }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|e^{iHt'/\hbar }a_{\gamma }^{+}a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2262c7708c2a09555a044ec5d564eb2f3fc2f134)
![{\displaystyle =-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}a_{\alpha }e^{-iHt/\hbar }|\psi _{o}^{N}\rangle \langle \psi _{o}^{N}|e^{iHt'/\hbar }a_{\gamma }^{+}a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle -{\frac {i}{\hbar }}\sum _{n\neq 0}^{}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}a_{\alpha }e^{-iHt/\hbar }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|e^{iHt'/\hbar }a_{\gamma }^{+}a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/991ecc241aa735b8d47466a95771fdd46a0628f0)
(Eq. 2)
![{\displaystyle =-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{o}^{N}\rangle \langle \psi _{o}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle -{\frac {i}{\hbar }}[\sum _{n\neq 0}^{}\theta (t-t')e^{i(E_{o}^{N}-E_{n}^{N})(t-t')/\hbar }\langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle +\sum _{n\neq 0}^{}\theta (t'-t)e^{i(E_{o}^{N}-E_{n}^{N})(t'-t)/\hbar }\langle \psi _{o}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{o}^{N}\rangle ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1693b984fe07c0b6b9401a26535ac3735fd692)
where the definition of the time-ordering operator in terms of step functions is used also. The so-called polarization propagator is defined by Eq. (2) which includes the excited states only: