We present a moment equation Rayleigh - Ritz (RR) variational theory for the ground-state energy. An extended Fourier transform space can be defined through the moment equation corresponding to a given Schrödinger Hamiltonian. Within this extended space, we can implement a variational ansatz with respects to configurations of the form in which the energy-dependent functions : (i) satisfy the momentum space Schrödinger equation; (ii) are uniquely prescribed; (iii) are (most likely) non-integrable; and (iv) yield the physical solution for physical E and values. On the basis of this representation, one can minimize the energy expectation value with respects to the E and missing moment variables, . The proposed approach is in sharp contrast to traditional configuration-space RR implementations in which the selection of a variational basis is not manifest a priori, particularly for problems of spatial dimension greater than one. The analysis of one- and two-dimension problems is presented.
We present a moment equation Rayleigh - Ritz (RR) variational theory for the ground-state energy. An extended Fourier transform space can be defined through the moment equation corresponding to a given Schrödinger Hamiltonian. Within this extended space, we can implement a variational ansatz with respects to configurations of the form in which the energy-dependent functions : (i) satisfy the momentum space Schrödinger equation; (ii) are uniquely prescribed; (iii) are (most likely) non-integrable; and (iv) yield the physical solution for physical E and values. On the basis of this representation, one can minimize the energy expectation value with respects to the E and missing moment variables, . The proposed approach is in sharp contrast to traditional configuration-space RR implementations in which the selection of a variational basis is not manifest a priori, particularly for problems of spatial dimension greater than one. The analysis of one- and two-dimension problems is presented.
Abstract We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.
from the Hamiltonian function üof classical mechanics by the operator substitution h d liri dqiwhere h is Planck's constant.For instance, for a hydrogen atom with fixed nucleus H is the sum of the kinetic energy of the one electron and the potential energy of the electron with respect to the nucleus :The Hamiltonian operator correspondingly involves the Laplacian operator in a term representing the kinetic energy of the electron and a scalar multiplier representing the potential energy:The wave equation is thenIn these equations £ is a number, the energy of the system, and *, the corresponding solution, is the wave function which describes the state of the system when it has that energy, x// is obviously a function of all the spatial coordinates needed to define a configuration of the system.In the case of a simple molecule like Cl 2 , for instance, \p would depend on 108 variables.It is this multiplicity of variables which makes it so hard for physicists to interest mathematicians in some problems they would like very much to have solved.Now not all the solutions of the wave equation are of interest, but only those which can be physically interpreted.This requirement of physical interpretability places certain auxiliary conditions on the functions \//, often merely that they be quadratically integrable or that they remain bounded.Between the physically interprétable wave functions and the stationary states in which the system may be found there is a one-to-one correspondence.Thus in many cases physically satisfactory solutions of the wave equation exist only for certain discrete values of E, and in such cases one finds that the physical system also can possess only these discrete energies.The physically satisfactory solutions \p of such an equation and the 194i]
We present a systematic study of the Rayleigh--Ritz variational method for quantum oscillators in the Segal--Bargmann space. We rigorously derive the normalizability condition ∣α∣<21 for generalized Gaussian trial functions ψ(z)=eαz2+βz through convergence analysis of Gaussian integrals in the complex plane. Applications to the harmonic oscillator demonstrate exact recovery of the ground state in Segal--Bargmann space when the trial family contains the true solution. For the quartic anharmonic oscillator (H^=−21∂x2+21x2+λx4), adaptive Gaussian ansätze in position space yield a cubic stationarity equation and perturbative energy expansions beyond first order, capturing anharmonic wavefunction narrowing. In contrast, monomial trial functions (ψn(z)=zn) in the Segal--Bargmann space -- while providing rigorous upper bounds En=n+21+43λ(2n2+2n+1) for excited states -- lack width adaptability and are limited to first-order accuracy for ground-state calculations. We further analyze displaced Gaussians and displaced monomials for asymmetric potentials (e.g., x3+x4), showing that displacement parameters are essential to capture parity breaking and stabilization effects (E0≈21+43μ−49λ2+⋯).
Abstract In this chapter we consider the time-independent Schrödinger equation for three spatial dimensions. Making use of the orbital angular momentum observables, including the Laplacian in spherical polar coordinates, and introducing an effective potential, we recast the three-dimensional Schrödinger as a one-dimensional differential equation for the reduced radial wave function. In this context we discuss conditions for the existence of bound states and an upper bound on the ground-state energy obtained from the Rayleigh-Ritz variational principle. As an example we then consider the bound-state solutions for the spherical potential well and discuss the conditions for it admitting a ground state. We then turn to the Coulomb potential representing the hydrogen atom and discuss methods for obtaining upper and lower bounds on the ground-state energy to explain the stability of matter. Finally, obtain the Laguerre-polynomial solutions of the radial Schrödinger equation for the hydrogen atom, determine their energies and discuss their properties
The constrained energy functional, defined by E0(a) =minSa{a〈Ψ‖Ĥ0‖Ψ〉a}, ai=a〈Ψ‖Âi‖Ψ〉a, i=1, ..., M, ‖Ψ〉a∈Sa, with a={ai} a preassigned set of expectation values generated by any state vector ‖Ψ〉a in the class Sa, is shown to satisfy a first order, but nonlinear, partial differential equation. The equation results by replacing the Lagrange multipliers introduced in an earlier paper by the corresponding partial derivatives of the to-be-found energy functional with respect to the constraints. Two alternative expressions for E0(a), one focusing upon the ground state eigenvalue of a modified Hamiltonian operator and the other upon the corresponding eigenvector, lead via standard Rayleigh–Schrödinger perturbation theory to two approximate partial differential equations for the energy functional. Both equations together with the imposed subsidiary conditions lead in turn to the same approximate E0(a). An example based on the linear harmonic oscillator illustrates the concepts presented in the formalism.
The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix 〈m1m2∣A∣p1p2〉. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix A. Hole-state energies are given self-consistently in terms of the matrix A, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.