For a nonrelativistic quantal particle moving with the angular momentum $l=0$ in a spherically symmetric, smooth potential well, which in the neighborhood of the center of force gives rise to an attractive Coulomb force, a formula for the probability density of the particle at the center of force is derived by means of certain phase-integral approximations of an arbitrary order. From this formula, which is expected to be very accurate, the non-relativistic approximation of the Fermi-Segr\`e formula for the probability density of finding an $s$ electron at the nucleus of an atom or ion is obtained. The limits of validity of the Fermi-Segr\`e formula in the nonrelativistic approximation are clarified by this derivation.For a nonrelativistic quantal particle moving with the angular momentum $l=0$ in a spherically symmetric, smooth potential well, which in the neighborhood of the center of force gives rise to an attractive Coulomb force, a formula for the probability density of the particle at the center of force is derived by means of certain phase-integral approximations of an arbitrary order. From this formula, which is expected to be very accurate, the non-relativistic approximation of the Fermi-Segr\`e formula for the probability density of finding an $s$ electron at the nucleus of an atom or ion is obtained. The limits of validity of the Fermi-Segr\`e formula in the nonrelativistic approximation are clarified by this derivation.

Physical Review ADec 1, 197224 citations

Classical limit of a relativistic quantum system

Hans Herlof Grelland

International Journal of Theoretical PhysicsMar 1, 19843 citations

Three-dimensional statistical reduction of the N -body Schrödinger equation for electrons with pairwise Coulomb interactions

Boyan Obreshkov

Based on a second-quantized representation of the nonrelativistic Hamiltonian of a system of $N$ electrons with pairwise Coulomb interactions, we demonstrate the exact statistical reduction of the $N$ -body problem to a three-dimensional Schr\"odinger equation for the motion of a single active electron with all other $N\ensuremath{-}1$ electrons acting as spectators. As a by-product, three-dimensional Schr\"odinger equations for the ground and excited states of two-electron atoms and ions are derived and the dynamical role of Pauli's exclusion principle is established. The classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ of the quantal all-electron equations is examined, and the Thomas-Fermi equation including the Amaldi correction is obtained.Based on a second-quantized representation of the nonrelativistic Hamiltonian of a system of $N$ electrons with pairwise Coulomb interactions, we demonstrate the exact statistical reduction of the $N$-body problem to a three-dimensional Schr\"odinger equation for the motion of a single active electron with all other $N\ensuremath{-}1$ electrons acting as spectators. As a by-product, three-dimensional Schr\"odinger equations for the ground and excited states of two-electron atoms and ions are derived and the dynamical role of Pauli's exclusion principle is established. The classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ of the quantal all-electron equations is examined, and the Thomas-Fermi equation including the Amaldi correction is obtained.

Physical Review ASep 3, 20080 citations

Semi-classical limit of relativistic quantum mechanics

Ladislav Kocis

PramanaJul 1, 20051 citation

Classical Limit of Quantum Physics

A. O. Bolivar

The frontiers collectionJan 1, 20041 citation

For a nonrelativistic quantal particle moving with the angular momentum

l=0

in a spherically symmetric, smooth potential well, which in the neighborhood of the center of force gives rise to an attractive Coulomb force, a formula for the probability density of the particle at the center of force is derived by means of certain phase-integral approximations of an arbitrary order. From this formula, which is expected to be very accurate, the non-relativistic approximation of the Fermi-Segr\`e formula for the probability density of finding an

s

electron at the nucleus of an atom or ion is obtained. The limits of validity of the Fermi-Segr\`e formula in the nonrelativistic approximation are clarified by this derivation.

Based on a second-quantized representation of the nonrelativistic Hamiltonian of a system of

N

electrons with pairwise Coulomb interactions, we demonstrate the exact statistical reduction of the

N

-body problem to a three-dimensional Schr\"odinger equation for the motion of a single active electron with all other $N\ensuremath{-}1$ electrons acting as spectators. As a by-product, three-dimensional Schr\"odinger equations for the ground and excited states of two-electron atoms and ions are derived and the dynamical role of Pauli's exclusion principle is established. The classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ of the quantal all-electron equations is examined, and the Thomas-Fermi equation including the Amaldi correction is obtained.