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

How are quantum eigenfunctions of hydrogen atom related to its classical elliptic orbits?

Yi-Xuan 艺轩 Yin 殷, Tiantian Wang, Biao Wu

Abstract We show that a highly-excited energy eigenfunction ψ nlm ( r ) of hydrogen atom can be approximated as an equal-weight superposition of classical elliptic orbits with energy E n and angular momentum L = l ( l + 1 ) ħ , and z component of angular momentum L z = mħ . This correspondence is established by comparing the quantum probability distribution | ψ nlm ( r )| 2 and the classical probability distribution p c ( r ) of an ensemble of such orbits. This finding illustrates a general principle: in the semi-classical limit, an energy eigenstate of a quantum system is in general reduced to a collection of classical orbits, rather than a single classical orbit.Abstract We show that a highly-excited energy eigenfunction ψ nlm ( r ) of hydrogen atom can be approximated as an equal-weight superposition of classical elliptic orbits with energy E n and angular momentum L = l ( l + 1 ) ħ , and z component of angular momentum L z = mħ . This correspondence is established by comparing the quantum probability distribution | ψ nlm ( r )| 2 and the classical probability distribution p c ( r ) of an ensemble of such orbits. This finding illustrates a general principle: in the semi-classical limit, an energy eigenstate of a quantum system is in general reduced to a collection of classical orbits, rather than a single classical orbit.

arXiv:2411.18890Chinese Physics BJan 4, 20260 citations

Classical-field model of the hydrogen atom

Sergey A. Rashkovskiy

arXiv:1602.04090Indian Journal of PhysicsFeb 25, 201718 citations

Stark states and O(4) symmetry of hydrogenic atoms

J. W. B. Hughes

The O(4) symmentry of hydrogenic atoms is used to interpret angular momentum nlm and Stark [vertical line] n 1 n 2 m eigenstates as different realizations of irreducible representations of the O(4) group Differential expressions are obtained for [vertical line] n 1 n 2 m shift operators, and for the [vertical line] nlm shift operator O - 1 which changes l and m by -1 These are used to obtain the formula < n 1 n 2 m | nlm '> = (-1) 2 n 2 +| m |- m 2 C [1/2( n -1), 1/2( n -1), l , 1/2( m + n 2 - n 1 ), 1/2( m + n 1 - n 2 ), m '] for the transformation matrix elements connecting the two sets of states where, in the notation of Rose, C ( J 1 J 2 J , m 1 m 2 m ) is the angular momentum Clebsch-Gordan coefficientThe O(4) symmentry of hydrogenic atoms is used to interpret angular momentum nlm and Stark [vertical line] n 1 n 2 m eigenstates as different realizations of irreducible representations of the O(4) group Differential expressions are obtained for [vertical line] n 1 n 2 m shift operators, and for the [vertical line] nlm shift operator O - 1 which changes l and m by -1 These are used to obtain the formula < n 1 n 2 m | nlm '> = (-1) 2 n 2 +| m |- m 2 C [1/2( n -1), 1/2( n -1), l , 1/2( m + n 2 - n 1 ), 1/2( m + n 1 - n 2 ), m '] for the transformation matrix elements connecting the two sets of states where, in the notation of Rose, C ( J 1 J 2 J , m 1 m 2 m ) is the angular momentum Clebsch-Gordan coefficient

Proceedings of the Physical SocietyAug 1, 196791 citations

Wave Functions for the Hydrogen-like Atom

WORLD SCIENTIFIC eBooksJan 1, 20260 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.