The classical form factor is deduced from exact correspondence with a phase-space representation of the quantal form factor. Analytical expressions are provided for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}},$ $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}$, and $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ transitions in hydrogenic systems and for $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ in the one-dimensional harmonic oscillator. An efficient procedure for calculation of quantal form factors as analytical functions of momentum transfer, for arbitrary quantum numbers, is presented. The classical approach has the ability to explain quite succinctly interesting trends and various important aspects which remain hidden within the quantal treatment of form factors. The classical-quantal comparison ranges from being qualitatively good for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}$ transitions to close agreement for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}$ and $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ transitions. Excellent agreement is obtained for the integrated form factor for all transitions.The classical form factor is deduced from exact correspondence with a phase-space representation of the quantal form factor. Analytical expressions are provided for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}},$ $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}$, and $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ transitions in hydrogenic systems and for $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ in the one-dimensional harmonic oscillator. An efficient procedure for calculation of quantal form factors as analytical functions of momentum transfer, for arbitrary quantum numbers, is presented. The classical approach has the ability to explain quite succinctly interesting trends and various important aspects which remain hidden within the quantal treatment of form factors. The classical-quantal comparison ranges from being qualitatively good for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}$ transitions to close agreement for $n\stackrel{\ensuremath{\rightarrow}}{l}{n}^{\ensuremath{'}}$ and $\stackrel{\ensuremath{\rightarrow}}{n}{n}^{\ensuremath{'}}$ transitions. Excellent agreement is obtained for the integrated form factor for all transitions.

Physical Review AAug 1, 19999 citations

Classical and quantal atomic form factors for nlm→n′l′m transitions

M. Raymond Flannery, D. Vrinceanu

An analytical expression for the classical form factor or impulsive probability ${P}_{\mathrm{if}}(\mathbf{q})$ for $\mathrm{nl}\stackrel{\ensuremath{\rightarrow}}{m}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}m$ transitions is derived directly from the ``phase-space distribution'' method [Phys. Rev. A 60, 1053 (1999)] and is compared with quantal results. Exact universal scaling laws are derived for the classical probability for any $\stackrel{\ensuremath{\rightarrow}}{i}f$ transition. As n is increased, convergence of the quantal to classical results is obtained and it becomes even more rapid upon averaging in succession over the m and then the $\mathit{l}$ substates. The classical results reveal the basic reason for the underlying structure in the variation of ${P}_{\mathrm{if}}$ with momentum transfer $\mathbf{q}.$ Classical form factors can operate as an effective averaged version of the exact quantal counterpart.An analytical expression for the classical form factor or impulsive probability ${P}_{\mathrm{if}}(\mathbf{q})$ for $\mathrm{nl}\stackrel{\ensuremath{\rightarrow}}{m}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}m$ transitions is derived directly from the ``phase-space distribution'' method [Phys. Rev. A 60, 1053 (1999)] and is compared with quantal results. Exact universal scaling laws are derived for the classical probability for any $\stackrel{\ensuremath{\rightarrow}}{i}f$ transition. As n is increased, convergence of the quantal to classical results is obtained and it becomes even more rapid upon averaging in succession over the m and then the $\mathit{l}$ substates. The classical results reveal the basic reason for the underlying structure in the variation of ${P}_{\mathrm{if}}$ with momentum transfer $\mathbf{q}.$ Classical form factors can operate as an effective averaged version of the exact quantal counterpart.

Physical Review AJan 4, 20025 citations

An analytical expression for the classical form factor or impulsive probability

{P}_{\mathrm{if}}(\mathbf{q})

for $\mathrm{nl}\stackrel{\ensuremath{\rightarrow}}{m}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}m$ transitions is derived directly from the ``phase-space distribution'' method [Phys. Rev. A 60, 1053 (1999)] and is compared with quantal results. Exact universal scaling laws are derived for the classical probability for any $\stackrel{\ensuremath{\rightarrow}}{i}f$ transition. As n is increased, convergence of the quantal to classical results is obtained and it becomes even more rapid upon averaging in succession over the m and then the

\mathit{l}

substates. The classical results reveal the basic reason for the underlying structure in the variation of

{P}_{\mathrm{if}}

with momentum transfer

\mathbf{q}.

Classical form factors can operate as an effective averaged version of the exact quantal counterpart.