Variational Principle for a Particle in a Box

Particle in a box is one of the applications of the Schrodinger equation. Schrodinger equation gives wave function which is used to determine the probability and expectation value of particle in a box. It is also can explain the energy levels of the particle. This study aimed to determine the probability, expectation values, and energy levels of a particle in a three-dimensional box with variations in main quantum numbers in each coordinate axis. The magnitude of the wave function and energy levels are influenced by the main quantum number and the width of the box. The results show that variations in the main quantum number influence the probability of particle in the three-dimensional box except for the width of the box and L the probability of particle showing the same value in all variations of the main quantum number. Variations in the main quantum number also influence the expectation value of particle in a three-dimensional box except for the width of the box L the expectation value shows the same value in all variations of the main quantum number. All variations of the main quantum number also influence the magnitude of the energy level of the particle.

Phenomenological two-nucleon potentials and the deuteron

The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.

An approximate light-cone wave function for the pion effective quark-antiquark Fock sector corresponding to a small value of the renormalization group parameter is presented. The approximate wave function is motivated by the LF-holography and the quadratic confinement potential in the front form of Hamiltonian dynamics, which is in harmony with the linear confining potential in the instant form. The pion radius, decay constant and form-factor are also presented.

The spectrum of an electron in a dielectric junction is analysed in terms of one-dimensional box-confinement with width 2a and a potential due to images in the two parallel planes. It tends to the spectrum of a particle in a box for a→0, and to the Coulombic spectrum with parity degeneracy for a→∞. Based on some general properties, simple expressions are obtained for the energies of all the states. Model wavefunctions are developed for the first four states which provide a physical insight into their structure.

The time-dependent variational principle in the stationary action principle form is formulated with constraint conditions for parametrized wave functions. The constraint conditions are classified into the first and the second classes as in Dirac's constrained classical mechanics based on the commutability of operators. If the local bases in the parametrized wave functions construct a complete basis set, the classification of constraints with the commutator becomes equivalent to that with the complex generalized Poisson brackets (CGPB). However, in approximate wave functions with limited variational parameters, the equivalence between the CGPB and the commutator of operators is lost. The first-class constraints and the constants of motion classified with the commutator should be reconsidered as the pseudo-second-class constraints in the approximate wave functions.

Cluster perturbation (CP) theory was developed in Paper I [F. Pawłowski et al., J. Chem. Phys. 150, 134108 (2019)] for a coupled cluster (CC) target state and is extended in this paper to comprehend a cluster linear (CL) target state, for which the embedding of a CC parent state in the target excitation space is described using a linear parametrization. The theory is developed for determining the energy and molecular properties for a CL state. When CP theory is applied to a CL target state, a series of corrections is determined in orders of the CC parent-state similarity-transformed fluctuation potential, where the zeroth-order term is the energy or molecular property of the CC parent state and where the series formally converges to the energy or molecular property of the CL target state. The determination of energies and molecular properties is simpler for a CL state than for a CC state because the CL state is linearly parametrized. The amplitude equations are quadratic for a CL target state, while quartic for a CC target state, and molecular property expressions for a CL target state have the same simple structure as for a configuration interaction state. The linear parametrization introduces non-size-extensive contributions in the energy and molecular property expressions. However, since the linear parametrization describes the embedding of the CC parent state in the target excitation space, the energy and molecular properties for a CL state are weakly size-extensive. For the energy, weak size-extensivity means that non-size-extensive contributions enter in sixth and higher orders in the CP energy series, whereas for molecular properties, weak size-extensivity means that non-size-extensive contributions enter in second and higher orders. Weak size-extensivity therefore has a little or vanishing effect on calculated energies or molecular properties. The determination of the CP energy and molecular property corrections does not require that amplitude or response equations are solved explicitly for the target state and it becomes computationally tractable to use low-order corrections from these series to obtain energies and molecular properties of CL target state quality. For three simple molecules, HF, N2, and CH2, the accuracy of the CL approach for ground-state energies is tested using a parent state including single and double excitations (i.e., the CC singles-and-doubles state, CCSD) and a target state that includes triple excitations. It is found that the size-extensive fifth-order CL energies deviate by less than 0.0001 hartree from the energies of a target CC that includes triple excitations (i.e., the CC singles-doubles-and-triples state, CCSDT). CP theory with a CL target state therefore becomes a very attractive replacement of standard CC theory for high-accuracy energy and molecular property calculations, in which triple and higher excitation levels are considered.

ADVERTISEMENT RETURN TO ISSUEPREVTech ReportNEXTEnergies and Wave Functions for Several One-Dimensional PotentialsRicardo Metz View Author Information Department of Chemistry, Lederle GRT, University of Massachusetts, Amherst, MA 01003-4510Cite this: J. Chem. Educ. 2004, 81, 1, 157Publication Date (Web):January 1, 2004Publication History Received3 August 2009Published online1 January 2004Published inissue 1 January 2004https://pubs.acs.org/doi/10.1021/ed081p157https://doi.org/10.1021/ed081p157product-reviewACS Publications. This publication is available under these Terms of Use. Request reuse permissions This publication is free to access through this site. Learn MoreArticle Views815Altmetric-Citations4LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail PDF (35 KB) Get e-AlertscloseSupporting Info (9)»Supporting Information Supporting Information SUBJECTS:Energy,Oscillation,Students,Wave function Get e-Alerts

A finite set of Slater-type orbitals is used as a basis to solve the Dirac equation for atoms in a Dirac-Fock-Slater potential. The energy eigenvalues thus obtained are in good agreement with the results from the numerical solution of the Dirac equation except that a spurious root always occurs for \ensuremath{\kappa}>0. It is shown that despite this spurious root the method still gives good approximate atomic energy eigenvalues and wave functions. This implies that the finite-basis-set expansion method can be applied to relativistic atomic structure and atomic collision calculations.

Certain aspects of Rayleigh-Schr\"odinger (RS) perturbation theory and of the variational principle are combined in a new perturbationally oriented variational approach for obtaining approximate solutions to Schr\"odinger's time-independent equation. This perturbational-variational (PV) procedure is applicable for all perturbed Hamiltonian operators which can be expanded as $H(\ensuremath{\lambda})=\ensuremath{\Sigma}{H}_{t}{\ensuremath{\lambda}}^{t}$, where $\ensuremath{\lambda}$ is an external parameter governing the strength of the perturbing terms; in particular, the theory is developed here for the important special form, $H={H}_{0}+{H}_{1}\ensuremath{\lambda}$. It is shown how the application of the PV procedure to an arbitrary analytic approximate wave function yields the variational parameters embedded in the wave function, the wave function itself, the corresponding approximate energy, and other expectation values, all in the form of energetically optimized PV expansions in powers of $\ensuremath{\lambda}$ to any desired order. A variational wave-operator formalism is introduced to deal with the multivariant expansions required to form the PV expansions. The remainder theorem and the variational Hellmann-Feynman theorem are derived and their role in the PV methodology is discussed. The principal advantages of the new method are: (1) Unlike RS perturbation theory, there is no requirement that the exact solution of a simpler unperturbed system be known; (2) the explicit $\ensuremath{\lambda}$ dependence of the optimum quantities is analytically derived; (3) the numerical aspects of a given variational problem are simplified because the variational equations generated by the PV procedure are almost completely linearized in the variational parameters; and (4) the PV expansions form a natural framework for the systematic classification and evaluation of all perturbed variational wave functions in respect to their quality. The PV classification scheme, previously presented in bare outline, is rigorously derived. The insight obtained from PV theory is further illustrated by the derivation and application of the variational integral Hellmann-Feynman theorem, the virial theorem, and a theorem relating to the expectation value of the unperturbed potential. The a priori PV classification of perturbed variational wave functions is extended to the perturbed variational parameters embedded in the wave functions; it is shown that the anomalous PV expansions of some open-shell parameters and orbitals can be simply explained in terms of PV theory.

We introduce a Bloch-like basis in a C-component lowest Landau level fractional quantum Hall (FQH) effect, which entangles the real and internal degrees of freedom and preserves an N(x)×N(y) full lattice translational symmetry. We implement the Haldane pseudopotential Hamiltonians in this new basis. Their ground states are the model FQH wave functions, and our Bloch basis allows for a mutatis mutandis transcription of these model wave functions to the fractional Chern insulator of arbitrary Chern number C, obtaining wave functions different from all previous proposals. For C>1, our wave functions are related to color-dependent magnetic-flux inserted versions of Halperin and non-Abelian color-singlet states. We then provide large-size numerical results for both the C = 1 and C = 3 cases. This new approach leads to improved overlaps compared to previous proposals. We also discuss the adiabatic continuation from the fractional Chern insulator to the FQH in our Bloch basis, both from the energy and the entanglement spectrum perspectives.

The strong and singular interaction between nucleons, makes the nuclear many body theory very complicated. Still, nuclei exhibit simple and regular features which are simply described by the shell model. Wave functions of individual nucleons may be considered just as model wave functions which bear little resemblance to the real ones. There is, however, experimental evidence for the reality of single nucleon wave functions. There is a simple method of constructing such wave functions for valence nucleons. It is shown that this method can be improved by considering the polarization of the core by the valence nucleon. This gives rise to some rearrangement energy which affects the single valence nucleon energy within the nucleus.

The use of interparticle coordinates in atomic and molecular calculations is reviewed and emphasized. For states where the wave function is dependent only upon interparticle coordinates the Hamiltonian is a relatively simple expression. If the wave function is a linear combination of products of functions of single interparticle coordinates of the exponential power form the formulas for local energy are particularly simple.

Uniform WKB wave functions are developed for prolate spheroidal wave functions which arise when the Schrödinger equation is given in prolate spheroidal coordinates. A detailed numerical test is presented for angular spheroidal wave functions. It shows that the approximate uniform WKB wave functions are excellent in accuracy.

Cluster structure in nuclei