scholarly journals The energy of crystal lattices

1930 ◽  
Vol 127 (806) ◽  
pp. 689-703 ◽  
Keyword(s):  
Perturbation Theory ◽  
Energy State ◽  
Wave Functions ◽  
Exclusion Principle ◽  
Original Form ◽  
Spin Moment ◽  
Crystal Lattices ◽  
Initial Wave ◽  
Starting Point ◽  
Adiabatic Approach

In order to calculate the potential energy of a collection of a large number of atoms it is necessary to use the quantum mechanical perturbation theory. The choice of the initial wave functions with which the perturbation calculation is to be carried out, is equivalent to deciding what model of the system shall be taken as the starting point. In the case of crystals the model which involves the simplest assumptions is that in which the crystal is regarded initially as a large number of atoms in their lowest energy state, arranged in a lattice; the lattice constant being great. The perturbation theory is then applied to find how the energy of the system changes as the atoms are brought slowly together; the lattice retaining its original form. It is not necessarily true, however, that when the separation has been reduced to that actually occurring in a given crystal that the system of normal atoms, adiabatically brought together, will be identical with the crystal itself. Thus, for example, Hertzberg has shown that the deepest state of N 2 + does not arise from the adiabatic approach of a normal N atom, and a normal N + ion. When the atoms of the lattice are still well separated it is possible to calculate the mean first order energy using a method given by Heitler. The determinant of the secular equation can be reduced, as Wigner has shown, to a number of irreducible sub-determinants to each of which corresponds a particular term system, and of these, those which satisfy the exclusion principle determine a given total spin moment. The sum of all the energies belonging to one-term system is then given by summing along the diagonal of the corresponding sub-determinant. This procedure, however, assumes that the initial waver functions form an orthogonal set. In the case of a crystal the initial wave functions are to be taken as the product of the wave functions of the separate atoms, and these will only be even approximately orthogonal when the distance between nearest atoms is great.

scholarly journals Chiral perturbation theory for GR

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Kirill Krasnov ◽  
Yuri Shtanov
Keyword(s):  
Perturbation Theory ◽  
Chiral Perturbation ◽  
Perturbative Calculation ◽  
New Formalism ◽  
Yang Mills ◽  
First Order ◽  
Starting Point ◽  
Gauge Fixing ◽  
New Perturbation ◽  
Effective Description

Abstract We describe a new perturbation theory for General Relativity, with the chiral first-order Einstein-Cartan action as the starting point. Our main result is a new gauge-fixing procedure that eliminates the connection-to-connection propagator. All other known first-order formalisms have this propagator non-zero, which significantly increases the combinatorial complexity of any perturbative calculation. In contrast, in the absence of the connection-to-connection propagator, our formalism leads to an effective description in which only the metric (or tetrad) propagates, there are only cubic and quartic vertices, but some vertex legs are special in that they cannot be connected by the propagator. The new formalism is the gravity analog of the well-known and powerful chiral description of Yang-Mills theory.

scholarly journals Quantum square-well with logarithmic central spike

2018 ◽  
Vol 33 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Miloslav Znojil ◽  
Iveta Semorádová
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Closed Form ◽  
Wave Functions ◽  
Change Of Variables ◽  
State Dependent ◽  
Square Well ◽  
Strength Formula ◽  
Schrödinger Perturbation ◽  
Dependent Interaction

Singular repulsive barrier [Formula: see text] inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction [Formula: see text] in nonlinear Schrödinger equation. In the linearized case, Rayleigh–Schrödinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small [Formula: see text] or after an amendment of the unperturbed Hamiltonian. At any spike strength [Formula: see text], the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables [Formula: see text] which interchanges the roles of the asymptotic and central boundary conditions.

Atomic and Molecular Structure

2019 ◽  
pp. 323-358
Author(s):  
P.J.E. Peebles
Keyword(s):  
Molecular Structure ◽  
Ground State ◽  
Molecular Hydrogen ◽  
Energy State ◽  
Cosmic Ray ◽  
Pauli Exclusion Principle ◽  
Exclusion Principle ◽  
Distant Galaxy ◽  
Long Time ◽  
Main Approximation

This chapter assesses some applications drawn from atomic and molecular structure. It deals with the structures of the lighter atoms and the simplest molecule, molecular hydrogen. The main approximation method used here is the energy variational principle, which is a powerful technique for computing the low-lying energies of a system such as an atom or molecule. The chapter then introduces the Pauli exclusion principle, which governs the symmetry of the state vector for a system of identical particles such as electrons. Two general features of the exclusion principle are worth noting. First, although the spins make only a very weak contribution to the Hamiltonians for helium, the lowest energy state with spin one is above the spin zero ground state, which is a considerable difference. Second, an electron arriving as a cosmic ray particle from a distant galaxy has to have a wave function antisymmetric with respect to the local electrons, even though the new electron has been away from us for a long time.

scholarly journals Quarks im Lichte einer Idee von Pais

1980 ◽  
Vol 35 (2) ◽  
pp. 252-253
Author(s):  
Fritz Bopp
Keyword(s):  
Wave Equation ◽  
Infinite Sequence ◽  
Quantum States ◽  
Wave Functions ◽  
Exclusion Principle ◽  
Repulsive Forces

Abstract A wave equation of a kind proposed by Pais in 1953 describes a particle with an infinite sequence of quantum states, which belong to the symmetrical representations (λ, 0) of the group SU 3. Particles composed of such single ones are connected with the whole set of representations (λ, μ) of SU 3. The wave equation is compatible with an exclusion principle. Assuming that only particles with zero triality occur, all quarks and quarklike particles are excluded. Neither coulours, nor bags are needed, as we do not need repulsive forces to exclude Li-atoms with symmetrical wave functions.

scholarly journals Exploration of Wireless Sensor Network Based on Cloud Computing

2018 ◽  
Vol 14 (11) ◽  
pp. 16
Author(s):  
Qing Wan ◽  
Ying Wang
Keyword(s):  
Genetic Algorithm ◽  
Cloud Computing ◽  
Wireless Sensor Network ◽  
Sensor Network ◽  
Energy State ◽  
Wireless Sensor ◽  
Cloud Environment ◽  
Application Service ◽  
Operation And Maintenance ◽  
Starting Point

To realize the exploration of wireless sensor network (WSN) based on cloud computing, the application service of WSN is taken as the starting point, the resource advantage of the cloud platform is used, and a WSN service framework based on cloud environment is proposed. Based on this framework, the problems of data management and reconstruction, network coverage optimization and monitoring, and edge recognition of holes are solved. In view of the node deployment of WSN and coverage problem of operation and maintenance optimization, the genetic algorithm is used to adjust the dormancy and energy state of nodes, and a parallel genetic algorithm for covering optimization in the cloud environment is proposed. For the operation and maintenance requirements of WSN, a parallel data statistics method for network monitoring is proposed. The experimental results show that the parallel algorithm is greatly improved in terms of the accuracy and time efficiency.

APPROXIMATE MOLECULAR ORBITALS: III. THE 1sσ AND 2pπ STATES OF HeH++

1967 ◽  
Vol 45 (7) ◽  
pp. 2231-2238 ◽  
Author(s):  
M. Cohen ◽  
R. P. McEachran ◽  
Sheila D. McPhee
Keyword(s):  
Perturbation Theory ◽  
Molecular Orbitals ◽  
Wave Functions ◽  
Variational Techniques ◽  
Approximate Wave ◽  
Schrödinger Perturbation

A combination of Rayleigh–Schrödinger perturbation theory and variational techniques, previously used to calculate the wave functions of the lowest σ and π states of H2+ has been applied to the 1sσ and 2pπ states of HeH++. The accuracy of the resulting approximate wave functions is demonstrated by comparing a number of quantities calculated with them with the corresponding exact values.

KANTBP: A program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

2007 ◽  
Vol 177 (8) ◽  
pp. 649-675 ◽  
Author(s):  
O. Chuluunbaatar ◽  
A.A. Gusev ◽  
A.G. Abrashkevich ◽  
A. Amaya-Tapia ◽  
M.S. Kaschiev ◽  
...  
Keyword(s):  
Energy Levels ◽  
Wave Functions ◽  
Radial Wave ◽  
Reaction Matrix ◽  
Adiabatic Approach ◽  
Hyperspherical Adiabatic ◽  
Coupled Channel

scholarly journals On the electrodynamic self-energy of the electron

1948 ◽  
Vol 195 (1041) ◽  
pp. 163-173
Keyword(s):  
Perturbation Theory ◽  
Wave Equation ◽  
Quantum Electrodynamics ◽  
Energy State ◽  
Energy Eigenvalue ◽  
Expansion Method ◽  
Negative Energy ◽  
The Self ◽  
Variation Principle ◽  
Self Energy

The stationary-state wave equation for an electron at rest in a negative-energy state in interaction with only its own electromagnetic field is considered. Quantum electrodynamics, single-electron theory and a ‘cut-off’ procedure in momentum-space are used. Expressions in the form of expansions in powers of e 2 /hc are derived for the wave function ψ and the energy-eigenvalue E by a method which (unlike perturbation theory) is not based on the assumption that the self-energy is small. The convergence of the expansion for E is not proved rigorously but the first few terms are shown to decrease rapidly. For low cut-off frequencies K 0 the expression for E behaves as the equivalent perturbation expression but for large K 0 it behaves as — J(e 2 /hc) hK0. The variation principle is applied to an approximation (obtained from the expansion method) for r/r, and it is proved rigorously that for large K 0 the self-energy is algebraically less than or equal to —J(e 2 /hc) hK 0 . Hence, if the electron wave-equation is considered as the limiting case of the ‘cut-off’ equation as K 0 ->ao, it is established that the divergences obtained are not merely due to improper use of perturbation theory and that the self-energy is indeed infinite.

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