Knowledge Bank

We consider $Na$ identical atoms in a gas of volume V; each atom has a $Jw=0+$ ground state and L excited $1-$ levels. The 3 $NaL$ excited states are labelled $|RI\alpha \nu ⟩$ ; they have at $R―I$ partner $\alpha (\alpha =x,y,z)$ of the $\nu th$ $1-$ level, of excitation energy $E\nu$, and have at all other atomic positions unexcited atoms. The atoms are assumed not to overlap. The Hamiltonian is $H=Ho+Hint$, where $Ho$ is the sum of $Na$ one-atom terms and $Hint$ is the sum of $Na(Na-1)/2$ Coulomb Interaction terms. Here the Hamiltonian matrix $$\langle{R}{I}\alpha\nu|{H}| {R}{I}^{\prime}\alpha^{\prime} \nu^{\prime}$$ is the Kronecker product $G\times \epsilon$ of a 3 $Na$ by 3 $Na$ “geometrical” matrix G and an L by L “electronic” matrix $\epsilon$. Similarity transformation of H by L $\Gamma \times IT$, where $\Gamma$ is the matrix of eigenvectors of G ($\Sigma I\prime \alpha \prime GI\alpha ,I\prime \alpha \prime \Gamma I\prime \alpha \prime ,n=gn\Gamma I\alpha ,n$ and $IL=(\delta \nu \nu \prime$) is the L-dimensional unit matrix, reduces H to 3 $Na$ diagonal L by L blocks with zeros elsewhere. Furthermore, the $nth$ such block of H-W has matrix elements $(H-W)\nu \nu \prime =(E\nu -W)\delta \nu \nu \prime +gne2Z\nu oZ\nu o\prime rs3$, where $-eZ\nu o$ is the transition moment between the $vth$ excited and the ground state of any atom, and $rs$ (defined by $4\pi rs3/3=V/Na$) is a scale length introduced to make G and g dimensionless, The determinant of H-W, det(H-W), is $\Pi n\{(\Pi \nu \lambda n\nu (1+\Sigma \nu 1/\lambda n\nu )\}$, where $\lambda \nu =(E\nu -W)/(gne2Z\nu oZ\nu o/rs3$). The secular equation det $(H-W)=0$ has a simple graphical interpretation for the energy eigenvalues $Wnv$ and a surprisingly simple equation for the coefficients of the excited states. All the above can be carried through explicitly if the atomic positions $RI$ form a lattice; the states are then the “exciton states” of solid state physics. For the argon resonance line the shift is about 1 in $105$ at 1000 torr.