# Introducing the Random Phase Approximation Theory

## Abstract

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## 1. Introduction

“The development, frequent independent rediscovery and gradual appreciation of the Random Phase Approximation offers a useful lesson to theoretical physicist. First, it illustrates the splendid variety of ways that can be developed for saying the same thing. Second, it suggests the usefulness of learning different languages of theoretical physics and of attempting the reconciliation of seemingly different, but obviously related results."

## 2. Independent Particle Models

#### 2.1. Mean-Field Model

#### 2.2. Hartree–Fock Theory

#### 2.3. Density Functional Theory

- (a)
- Because of the bijective mapping$${\widehat{U}}_{ext}\u27fa|{\mathrm{\Psi}}_{0}\rangle \u27fa{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}.$$
- (b)
- Because of (a), every observable is also a functional of ${\rho}_{0}$. Specifically, this is true for the energy of the system$$E\left[{\rho}_{0}\right]=\langle \mathrm{\Psi}\left[{\rho}_{0}\right]|\widehat{H}|\mathrm{\Psi}\left[{\rho}_{0}\right]\rangle =F\left[{\rho}_{0}\right]+\int {d}^{3}r\phantom{\rule{0.166667em}{0ex}}{\widehat{U}}_{ext}\left(\mathbf{r}\right)\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}\left(\mathbf{r}\right),$$$$F\left[{\rho}_{0}\right]\equiv \langle \mathrm{\Psi}\left[{\rho}_{0}\right]|\left(\widehat{T}+\widehat{V}\right)|\mathrm{\Psi}\left[{\rho}_{0}\right]\rangle .$$
- (c)
- The variational principle implies that for each $\rho \ne {\rho}_{0}$ the following relation holds:$${E}_{0}\equiv E\left[{\rho}_{0}\right]<E\left[\rho \right].$$

#### 2.4. Excited States in the Independent Particle Model

## 3. RPA with the Equation of Motion Method

#### 3.1. Tamm–Dankoff Approximation

#### 3.2. Random Phase Approximation

#### 3.2.1. Limits of the TDA

#### 3.2.2. RPA Equations

#### 3.2.3. Properties of RPA Equations

- If $B=0$, we obtain the TDA equations.
- We take the complex conjugate of the above equations and obtain$$\left(\begin{array}{cc}A& B\\ & \\ {B}^{*}& {A}^{*}\end{array}\right)\left(\begin{array}{c}{Y}^{*\nu}\\ \\ {X}^{*\nu}\end{array}\right)=-{\omega}_{\nu}\left(\begin{array}{c}{Y}^{*\nu}\\ \\ -{X}^{*\nu}\end{array}\right).$$This indicates that RPA equations are satisfied by positive and negative eigenvalues with the same absolute value.
- Eigenvectors corresponding to different eigenvalues are orthogonal.$$\left(\begin{array}{cc}A& B\\ & \\ {B}^{*}& {A}^{*}\end{array}\right)\left(\begin{array}{c}{X}^{\nu}\\ \\ {Y}^{\nu}\end{array}\right)={\omega}_{\nu}\left(\begin{array}{c}{X}^{\nu}\\ \\ -{Y}^{\nu}\end{array}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\begin{array}{cc}A& B\\ & \\ {B}^{*}& {A}^{*}\end{array}\right)\left(\begin{array}{c}{X}^{\mu}\\ \\ {Y}^{\mu}\end{array}\right)={\omega}_{\mu}\left(\begin{array}{c}{X}^{\mu}\\ \\ -{Y}^{\mu}\end{array}\right).$$Let us calculate the hermitian conjugate of the second equation$$({{X}^{\mu}}^{+},{{Y}^{\mu}}^{+})\left(\begin{array}{cc}A& B\\ & \\ {B}^{*}& {A}^{*}\end{array}\right)=({{X}^{\mu}}^{+},-{Y}^{\mu}){\omega}_{\mu}.$$We multiply the first equation by $({{X}^{\mu}}^{+},{{Y}^{\mu}}^{+})$ on the left hand side, and the second equation on the right hand side by$$\left(\begin{array}{c}{X}^{\nu}\\ \\ -{Y}^{\nu}\end{array}\right),$$$$\begin{array}{ccc}\hfill ({{X}^{\mu}}^{+},{{Y}^{\mu}}^{+})\left(\begin{array}{cc}A& B\\ & \\ {B}^{*}& {A}^{*}\end{array}\right)\left(\begin{array}{c}{X}^{\nu}\\ \\ {Y}^{\nu}\end{array}\right)& =& {\omega}_{\nu}({{X}^{\mu}}^{+},{{Y}^{\mu}}^{+})\left(\begin{array}{c}{X}^{\nu}\\ \\ -{Y}^{\nu}\end{array}\right)\hfill \\ \hfill & \phantom{\rule{3.33333pt}{0ex}}& \\ \hfill ({{X}^{\mu}}^{+},{{Y}^{\mu}}^{+})\left(\begin{array}{cc}A& B\\ & \\ {B}^{*}& {A}^{*}\end{array}\right)\left(\begin{array}{c}{X}^{\nu}\\ \\ {Y}^{\nu}\end{array}\right)& =& {\omega}_{\mu}({{X}^{\mu}}^{+},-{Y}^{\mu})\left(\begin{array}{c}{X}^{\nu}\\ \\ {Y}^{\nu}\end{array}\right).\hfill \end{array}$$By subtracting the two equations, we have$$0=({\omega}_{\nu}-{\omega}_{\mu})({{X}^{\mu}}^{+}{X}^{\nu}-{{Y}^{\mu}}^{+}{Y}^{\nu}).$$Since we assumed ${\omega}_{\nu}\ne {\omega}_{\mu}$, we obtain$$({{X}^{\mu}}^{+}{X}^{\nu}-{{Y}^{\mu}}^{+}{Y}^{\nu})=0.$$
- The normalization between two excited states requires$$\begin{array}{ccc}\hfill {\delta}_{\nu {\nu}^{\prime}}& =& \langle \nu |{\nu}^{\prime}\rangle =\langle {\nu}_{0}|{\widehat{Q}}_{\nu}{\widehat{Q}}_{{\nu}^{\prime}}^{+}|{\nu}_{0}\rangle =\langle {\nu}_{0}|[{\widehat{Q}}_{\nu},{\widehat{Q}}_{{\nu}^{\prime}}^{+}]|{\nu}_{0}\rangle \to \phantom{\rule{0.277778em}{0ex}}\mathrm{QBA}\to \langle {\Phi}_{0}|[{\widehat{Q}}_{\nu},{\widehat{Q}}_{{\nu}^{\prime}}^{+}]|{\Phi}_{0}\rangle \hfill \\ & =& \sum _{mi}\left({X}_{mi}^{\nu}{X}_{mi}^{{\nu}^{\prime}}-{Y}_{mi}^{\nu}{Y}_{mi}^{{\nu}^{\prime}}\right),\hfill \end{array}$$

#### 3.2.4. Transition Probabilities in RPA

#### 3.2.5. Sum Rules

#### 3.2.6. RPA Ground State

## 4. RPA with Green Function

#### 4.1. Field Operators and Pictures

#### 4.2. Two-Body Green Function and RPA

#### Infinite Systems

## 5. RPA with Time-Dependent Hartree–Fock

## 6. Continuum RPA

## 7. Quasi-Particle RPA (QRPA)

- ${\widehat{\mathcal{H}}}_{0}$ is purely scalar,$${\widehat{\mathcal{H}}}_{0}=\sum _{k}\left[({\u03f5}_{k}-\lambda -{\mu}_{k})2{v}_{k}^{2}+{u}_{k}{v}_{k}\sum _{{k}^{\prime}}{\overline{V}}_{k,{k}^{\prime},k,{k}^{\prime}}{u}_{{k}^{\prime}}{v}_{{k}^{\prime}}\right].$$
- ${\widehat{\mathcal{H}}}_{11}$ depends on ${\widehat{\alpha}}_{k}^{+}{\widehat{\alpha}}_{k}$,$${\widehat{\mathcal{H}}}_{11}=\sum _{k}\left\{\left[{\u03f5}_{k}-\lambda -{\mu}_{k}\right]({u}_{k}^{2}-{v}_{k}^{2})+2{u}_{k}{v}_{k}{\Delta}_{k}\right\}{\widehat{\alpha}}_{k}^{+}{\widehat{\alpha}}_{k}.$$
- ${\widehat{\mathcal{H}}}_{22}$ depends on $\widehat{\mathbb{N}}[{\widehat{\alpha}}_{k}^{+}{\widehat{\alpha}}_{{k}^{\prime}}^{+}+{\widehat{\alpha}}_{k}{\widehat{\alpha}}_{{k}^{\prime}}]$.
- ${\widehat{H}}_{\mathrm{int}}={\widehat{H}}_{40}+{\widehat{H}}_{31}+{\widehat{H}}_{22}$, where
- ${\widehat{H}}_{40}$ depends on $[{\widehat{\alpha}}_{{k}_{1}}^{+}{\widehat{\alpha}}_{{k}_{2}}^{+}{\widehat{\alpha}}_{{k}_{3}}^{+}{\widehat{\alpha}}_{{k}_{4}}^{+}+h.c.]$,
- ${\widehat{H}}_{31}$ depends on $[{\widehat{\alpha}}_{{k}_{1}}^{+}{\widehat{\alpha}}_{{k}_{2}}^{+}{\widehat{\alpha}}_{{k}_{3}}^{+}{\widehat{\alpha}}_{{k}_{4}}+h.c.]$,
- and finally,$${\widehat{H}}_{22}=\frac{1}{2}\sum _{abcd}{V}_{abcd}^{\left(22\right)}{\widehat{\alpha}}_{{k}_{a}}^{+}{\widehat{\alpha}}_{{k}_{b}}^{+}{\widehat{\alpha}}_{{k}_{d}}{\widehat{\alpha}}_{{k}_{c}},$$$${V}_{abcd}^{\left(22\right)}=({u}_{a}{u}_{b}{u}_{c}{u}_{d}+{v}_{a}{v}_{b}{v}_{c}{v}_{d}+4{u}_{a}{v}_{b}{u}_{c}{v}_{d}){\overline{V}}_{abcd}.$$