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Physical Review A

Euler-Lagrange Optimization of Plane-Wave Factors

M. E. Riley, Thomas A. Green

Published Aug 1, 1971• 68 citations
Publisher

Abstract

The Euler-Lagrange variational method is applied to plane-wave factors as they appear in the parametric time-dependent Schr"odinger-equation model for atom-atom scattering. The resulting Euler-Lagrange equations provide a prescription for determining a general plane-wave-factor function of the electron coordinates and time.
DOI
10.1103/physreva.4.619

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The quantum mechanical, time-dependent wave equation in generalized coordinates is obtained by a slight modification of Schrödinger's original derivation. If the classical Hamiltonian for a system is known, the variational technique may be applied to the Hamilton-Jacobi equation to yield an Euler-Lagrange equation. This latter is then the wave equation for the system. The method, which is generally easy to apply, is illustrated by applications to a single particle in curvilinear coordinates and to a symmetric top molecule.
The previous time-dependent variation-perturbation theory, which is derived from the conventional time-dependent perturbation equations, contains two inconvenient features. When the perturbation becomes static, the wave function so obtained does not satisfy the time-dependent Schr\"odinger equation, and the perturbation equations cannot be reduced to the ordinary time-independent perturbation equations. To remove these difficulties, we introduce a perturbed time-dependent phase factor in deriving a new set of perturbation equations. We then construct the variation-perturbation functionals as in the previous theory; the perturbed wave functions are solved by minimizing these functionals. The same method is also applied to many-electron systems, using the coupled and uncoupled Hartree-Fock scheme.
The ground-state wave function for a quantum electron gas is approximated by $\ensuremath{\psi}=\ensuremath{\Phi}D$, where DDD is the ground-state ideal-gas wave function and $\ensuremath{\Phi}=\mathrm{exp}[\ensuremath{-}(\frac{1}{2})\ensuremath{\Sigma}{i<j}^{}u({r}_{\mathrm{ij}})]$. Three-body correlation functions in the energy are approximated so that the energy is a functional of only pair-correlation functions. The variational principle is applied by utilizing the functional derivative of the pair-correlation function with respect to u(r)u(r)u(r). The three- and four-body correlation functions in the functional derivative are approximated in a consistent manner. Use of the Kirkwood superposition approximation for the many-body correlation functions in the functional derivative gives unphysical results. A new approximation to the functional derivative is obtained by using the convolution approximation for the many-body terms. This gives a reasonable result for the asymptotic form of the wave function. The hypernetted-chain functional derivative combined with the convolution approximation for the energy gives the same asymptotic form. When the random-phase approximation to the energy is used with the Gaskell-Broyles-Sahlin-Carley functional derivative, the equation may be solved analytically, giving the same result obtained by Gaskell. The asymptotic form is the same as the two previous methods, namely, $u\ensuremath{\sim}(\frac{e}{\ensuremath{\hbar}})\frac{{(\frac{m}{\ensuremath{\pi}\ensuremath{\rho}})}^{\frac{1}{2}}}{r}$.
The time-dependent Schrodinger for an atom in a radiation field has a very special form, at least within the dipole approximation: The time-dependence of the Hamiltonian is contained entirely in a c-number which appears as a factor in the interaction, V(t), of the atom with the radiation field. In the velocity gauge this factor is the vector potential, A(t), of the field — we have V(t)= − (e/µc)A (t) • p, where e, µ, and p are the charge, reduced mass, and canonical momentum (in the center of mass frame) of the electron — while in the length gauge this factor is the electric field vector F(t) — we have V(t) = –eF(t) • x, where x is the position coordinate. To our knowledge, this factorization property has not been fully exploited in previous approaches to solving the timedependent Schrodinger equation1. As we show here, in an application to multiphoton ionization of atomic hydrogen, it is an extremely useful property; and although at first sight it would seem to be a unique feature of the atom-radiation system, this is not so, for, as Solov’ev2 has pointed out, the time-dependent Schrödinger equation for colliding atoms or ions can be transformed into a form in which a scaled time-dependence also appears only in c-number factors of the (transformed) interactions.