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The Journal of Chemical Physics

Atomic orbitals of the nonrelativistic hydrogen atom in a four-dimensional Riemann space through the path integral formalism

Horacio Grinberg, Julio Marañón, H. Vucetich

Published Jan 15, 1983• 15 citations
Publisher

Abstract

The Kustaanheimo–Stiefel transformation together with the well-known expansion of the kernel of an isotropic harmonic oscillator is used to generate the atomic orbitals of the nonrelativistic hydrogen atom in a four-dimensional Riemann space through the path integral formalism. Group theoretical implications of the present problem are briefly discussed.

DOI
10.1063/1.444784

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COHERENT STATES OF HYDROGEN ATOM

Xu Bo-wei, ZENG QI, 上海交通大学物理系,上海,200030

From Kustannheimo-Stiefel transformation, the hygrogen problem in quantum mechanics can be transformed to that of a four dimensional isotropic harmonic oscillator with constrain. The coherent states are defined, for which the expectation values of the position and the momentum are shown to give the classical Kepler orbits. The uncertainty relations for the coherent states are discussed as well.

Acta Physica Sinica•Jan 1, 1991•2 citations
The Kustaanheimo–Stiefel transformation together with the well-known expansion of the kernel of an isotropic harmonic oscillator is used to generate the atomic orbitals of the nonrelativistic hydrogen atom in a four-dimensional Riemann space through the path integral formalism. Group theoretical implications of the present problem are briefly discussed.
A coordinate transformation which exhibits the rotational invariance of the hydrogen atom in four-dimensional Hilbert space is introduced. The coordinates are shown to be directly related to the spherical polar and parabolic coordinates in position space. With the use of the transformation, the Schr\"odinger equation for the hydrogen atom left-multiplied by 4r4r4r is transformed into one for a four-dimensional harmonic oscillator. Solutions are obtained and related to the hydrogenic wave functions. Group-theoretical implications of the transformation and its application to the hydrogen Stark problem are briefly discussed.
A research note connecting the shapes of atomic orbitals to four-dimensional geometry. The hydrogen atom possesses a hidden SO(4) symmetry (Pauli 1926) arising from conservation of the Runge-Lenz vector. In momentum space, hydrogen wavefunctions are exactly the spherical harmonics on the 3-sphere S³ (Fock 1935 stereographic projection). The familiar d and f orbital shapes are therefore 3D equatorial shadows of 4D hyperspherical harmonics. Covers: the Runge-Lenz vector and SO(4) Lie algebra; the Fock stereographic projection theorem; the Kustaanheimo-Stiefel transform (3D hydrogen ≡ 4D harmonic oscillator, exact); the SO(4,2) conformal group encoding the full hydrogen spectrum in a single representation; n-sphere generalization table; and open research directions including an interactive visualization paper (Am. J. Phys. target), angular-momentum dependence of AMEM matrix elements, and lanthanide crystal field analysis in the S³ harmonic basis.