Anti-hermitian contracted schrödinger equation: direct determination of the two-electron reduced density matrices of many-electron molecules

Two-electron reduced density matrices (2-RDMs) of many-electron molecules are directly determined without calculation of their wave functions by solving the anti-Hermitian contracted Schr?dinger equation. Approximation of the 3-RDM in the anti-Hermitian contracted Schr?dinger equation by a corrected cumulant expansion [Mazziotti, Phys. Rev. A 60, 3618 (1999)] permits the direct calculation of the energy and 2-RDM with many high-order correlation effects included. The method is illustrated for the molecules BeH2, H2O, NH3, CH4, and CO as well as the dissociation of BH. Correlation energies are obtained within 95%-100% of full-configuration interaction, and 2-RDMs very nearly satisfy known N-representability conditions.     

Deducing the Schrödinger equation from minimum χ2

The most unbiased probabilistic model for the possible values of a characteristic of a quantum system subject to the constraints represented by some known mean values characterizes the system in a steady-state condition. We suppose that random fluctuations alter such a steady-state condition. The probability distribution of the possible deviations from the steady-state condition is estimated by minimizing Pearson's ² subject to the mean fluctuations available. The optimum Pearson function ^* may be interpreted as the wave function of the system and in the case of the harmonic oscillator, the free particle in a box, and the hydrogen atom, the prediction based on it is compatible with that provided by the solution of the corresponding Schrödinger equations.     

Lagrangian formulation of the Schrödinger equation

We recast the Schrödinger equation in a new Lagrangian formulation. The equation is —i?dψ (x,t)/dt = Lψ (x,t), whereL is the Lagrangian operator. Expressions forL and ford/dt — ⊥ are derived in terms of coordinate and momentum operators.     

A canonical dilation of the Schrödinger equation

In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions.     

Statistical mechanics of the nonlinear Schrödinger equation

We investigate the statistical mechanics of a complex fieldø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian $$H(\phi ) = \int_\Omega {[\frac{1}{2}|\nabla \phi |^2 - (1/p) |\phi |^p ] dx}$$ is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, whenΩ is the circle and theL ² norm of the field (which is conserved by the dynamics) is bounded byN, the Gibbs measureυ obtained is absolutely continuous with respect to Wiener measure and normalizable if and only ifp andN are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, asN and the temperature are varied.     

The second hyperpolarizability of systems described by the space-fractional Schrödinger equation

The static second hyperpolarizability is derived from the space-fractional Schrödinger equation in the particle-centric view. The Thomas–Reiche–Kuhn sum rule matrix elements and the three-level ansatz determines the maximum second hyperpolarizability for a space-fractional quantum system. The total oscillator strength is shown to decrease as the space-fractional parameter α decreases, which reduces the optical response of a quantum system in the presence of an external field. This damped response is caused by the wavefunction dependent position and momentum commutation relation. Although the maximum response is damped, we show that the one-dimensional quantum harmonic oscillator is no longer a linear system for $\alpha \ne 1$, where the second hyperpolarizability becomes negative before ultimately damping to zero at the lower fractional limit of $\alpha \to 1/2$.     

Rapidly decaying solutions of the nonlinear Schrödinger equation

We consider global solutions of the nonlinear Schrödinger equation

Analytic presentation of a solution of the Schrödinger equation

High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into the nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.     

On the power series solution of the Schrödinger equation

A numerical method for solving the Schrödinger equation for a one-dimensional potential expressed as a function which increases in both directions away from its minimum is proposed. The basic assumption relies on the asymptotic properties of the solution. We exemplify the method calculating energies and expectation values for the quartic anharmonic oscillator.     

Local structure of the self-focusing singularity of the nonlinear Schrödinger equation

For the cubic Schrödinger equation in two dimensions we construct a family of singular solutions by perturbing slightly the dimension d = 2 tod > 2.     

Bounds on the density of states of random Schrödinger operators

Bounds are obtained on the unintegrated density of states ρ(E) of random Schrödinger operatorsH=?Δ + V acting onL ²(? ^d ) orl ²(? ^d ). In both cases the random potential is $$V: = \sum\limits_{y \in \mathbb{Z}^d } {V_y \chi (\Lambda (y))}$$ in which the \(\left\{ {V_y } \right\}_{y \in \mathbb{Z}^d }\) areIID random variables with densityf. The χ denotes indicator function, and in the continuum case the \(\left\{ {\Lambda (y)} \right\}_{y \in \mathbb{Z}^d }\) are cells of unit dimensions centered ony∈? ^d . In the finite-difference case Λ(y) denotes the sitey∈? ^d itself. Under the assumptionf ∈ L ₀ ^1+? (?) it is proven that in the finitedifference casep ∈ L ^∞(?), and that in thed= 1 continuum casep ∈ L _loc ^∞ (?).     

On the nonlinear Schrödinger limit of the Korteweg-de Vries equation

Using the multiscale approach of Zakharov and Kuznetsov it is shown that the nonlinear Schrödinger periodic scattering data is related to the Korteweg-de Vries periodic scattering data via an average over the Korteweg-de Vries carrier oscillation. This allows a complete elucidation of the physical meaning of the nonlinear Schrödinger scattering data, conservation laws, theta function solutions and reality constraint.     

L 2-exponential lower bounds to solutions of the Schrödinger equation

We study decay properties of solutions of the Schrödinger equation (–+V)=E. Typical of our results is one which shows that ifV=o(|x|^–1/2) at infinity or ifV is a homogeneousN-body potential (for example atomic or molecular), then ifE<0 and . We also construct examples to show that previous essential spectrum-dependent upper bounds can be far from optimal if is not the ground state.Research in partial fulfillment of the requirements for a Ph.D. degree at the University of VirginiaPartially supported by NSF grant MCS-81-01665Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Projekt Nr. 4240     

Invariant measures for the2D-defocusing nonlinear Schrödinger equation

Consider the2D defocusing cubic NLSiu _t+u–u|u|²=0 with Hamiltonian . It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing ||⁴ by ||⁴ :, is an invariant measure for the appropriately modified equationiu _t + u‒ [u|u ²–2(|u|² dx)u]=0. There is a well defined flow on thesupport of the measure. In fact, it is shown that for almost all data the solutionu, u(0)=, satisfiesu(t)–e ^it C _Hs (), for somes>0. First a result local in time is established and next measure invariance considerations are used to extend the local result to a global one (cf. [B2]).     

Schrödinger equation in the drift theory of cold plasma

A two-dimensional flow of an ideal neutral plasma across a magnetic field B is considered. The magnetic field is frozen in the plasma and is proportional to the plasma density n: Bn. All ions are assumed to have the same magnetic moment and, correspondingly, mechanical moment l. It is shown that the magnetic moment is doubled due to the drift motion. The equations of plasma hydrodynamics, to which terms proportional to l² have been added, are investigated within the framework of drift theory. The forces are due to the additional pressure of the drift velocity and are proportional to the Bohm potential . The equations derived by the Madelung transformation (transition to the function: ) are reduced to the Schrödinger cubic equation, which yields a new type of dynamics. It is shown that solitons, or nonspreading wave packets, which correspond to magnetosound waves in linear theory, and steady states can occur in the plasma described.Bauman State Technical University, Moscow. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 11, pp. 1133–1145, November, 1995.     

Exact solutions of the Schrödinger equation for zero energy

Some new exact solutions of the Schrödinger equation for zero energy are presented for certain nontrivial model potentials. Exact expressions for the different scattering lengths are derived and their differences and similarities are worked out. In particular, the respective distributions of the zeros and poles of the scattering lengths are characterized in detail.     

Quantization of the potential amplitude in the one-dimensional Schrödinger equation

A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ?_n(x) and eigenvalues α_n corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ?_n(x) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.