DickeModel.PhaseSpaces
This module provides formulas for the cannonical transformation $(Q,P)\mapsto (\theta,\phi)$ we use throughout this package.
There is an unfortunate problem between the characters $\phi$ (\phi) and $\varphi$ (\varphi): some fonts render one as the other. To avoid errors, the functions below treat the two characters indistinctly. In the references, the glyph $\phi$ is used.
DickeModel.PhaseSpaces.P_of_θφ — MethodSee P_of_θϕ.
DickeModel.PhaseSpaces.P_of_θϕ — MethodDickeModel.PhaseSpaces.Q_of_θφ — MethodSee Q_of_θϕ.
DickeModel.PhaseSpaces.Q_of_θϕ — MethodDickeModel.PhaseSpaces.arc_between_QP — Methodfunction arc_between_QP(Q1,P1,Q2,P2)Returns
arc_between_θϕ(θ_of_QP(Q1,P1),ϕ_of_QP(Q1,P1),θ_of_QP(Q2,P2),ϕ_of_QP(Q2,P2)).DickeModel.PhaseSpaces.arc_between_θφ — MethodSee arc_between_θϕ.
DickeModel.PhaseSpaces.arc_between_θϕ — Methodfunction arc_between_θϕ(θ1,φ1,θ2,φ2)Returns
\[ \Theta = \arccos(\cos(\theta_1)\cos(\theta_2)+ \sin(\theta_1)\sin(\theta_2)\cos(\phi_1 - \phi_2)).\]
DickeModel.PhaseSpaces.jx — Methodfunction jx(Q,P)Returns
\[ j_x = \sin(\theta(Q,P))\cos(\phi(Q,P)).\]
DickeModel.PhaseSpaces.jy — Methodfunction jy(Q,P)Returns
\[ j_y = \sin(\theta(Q,P))\sin(\phi(Q,P)).\]
DickeModel.PhaseSpaces.jz — Methodfunction jz(Q,P)Returns
\[ j_z = - \cos(\theta(Q,P)).\]
where $\text{arctan2}$ is the 2-argument arctangent.
DickeModel.PhaseSpaces.θ_of_QP — Methodfunction θ_of_QP(Q,P)Returns
\[ \theta =- \arccos\left(1 - \frac{Q^2 + P^2}{2}\right).\]
DickeModel.PhaseSpaces.φ_of_QP — MethodSee ϕ_of_QP.
DickeModel.PhaseSpaces.ϕ_of_QP — Methodfunction ϕ_of_QP(Q,P)Returns
\[ \phi = \text{arctan2}(-P,Q)\in [0,2 \pi ],\]
where $\text{arctan2}$ is the 2-argument arctangent.
Also φ_of_QP(Q,P).