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Eigenfunction

The last identity follows from the orthogonality property of eigenfunctions and the assumption of nomralization. The right-hand side in the final result is simply equal to the sum over all eigenvalues of the operator (possible results of the measurement) multiplied by the respective probabilities. Hence, an important corollary to the fiftli postulate is established ... [Pg.11]

This provides a recipe for calculating the average value of the system property associated with the quantum-mechanical operator A, for a specific but arbitrary choice of the wavefiinction T, notably those choices which are not eigenfunctions of A. [Pg.11]

The fifth postulate and its corollary are extremely important concepts. Unlike classical mechanics, where everything can in principle be known with precision, one can generally talk only about the probabilities associated with each member of a set of possible outcomes in quantum mechanics. By making a measurement of the quantity A, all that can be said with certainty is that one of the eigenvalues of /4 will be observed, and its probability can be calculated precisely. However, if it happens that the wavefiinction corresponds to one of the eigenfunctions of the operator A, then and only then is the outcome of the experiment certain the measured value of A will be the corresponding eigenvalue. [Pg.11]

In a more favourable case, the wavefiinction ]i might indeed correspond to an eigenfiinction of one of the operators. If = //, then a measurement of A necessarily yields and this is an unambiguous result. Wliat can be said about the measurement of B in this case It has already been said that the eigenfiinctions of two commuting operators are identical, but here the pertinent issue concerns eigenfunctions of two operators that do not conmuite. Suppose / is an eigenfiinction of A. Then, it must be true that... [Pg.15]

Note that h is simply the diagonal matrix of zeroth-order eigenvalues In the following, it will be assumed that the zeroth-order eigenfunction a reasonably good approximation to the exact ground-state wavefiinction (meaning that Xfi , and h and v will be written in the compact representations... [Pg.47]

The off-diagonal elements in this representation of h and v are the zero vector of lengtii (for h) and matrix elements which couple the zeroth-order ground-state eigenfunction members of the set q (for v) ... [Pg.47]

In order to solve (equation A1.4.7) we do not have to choose the basis fiinctions to be eigenfunctions of F and F, but there are obvious advantages in doing so ... [Pg.139]

To explain the vanishing integral nile we first have to explain how we detennine the synnnetry of a product. G fold degenerate state of energy and synnnetry T, with eigenfunctions - / -fold degen... [Pg.158]

The ability to assign a group of resonance states, as required for mode-specific decomposition, implies that the complete Hamiltonian for these states is well approxmiated by a zero-order Hamiltonian with eigenfunctions [M]. The ( ). are product fiinctions of a zero-order orthogonal basis for the reactant molecule and the quantity m. represents the quantum numbers defining ( ).. The wavefimctions / for the compound state resonances are given by... [Pg.1030]

Balint-Kurti G G, Dixon R N and Marston C C 1990 The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions J. Chem. See. Faraday Trans. 86 1741... [Pg.2326]

Neuhauser D 1990 Bound state eigenfunctions from wave packets—time -> energy resolution J. Chem. Phys. 932611... [Pg.2327]

The effective nuclear kinetic energy operator due to the vector potential is formulated by multiplying the adiabatic eigenfunction of the system, t t(/ , r) with the HLH phase exp(i/2ai ctan(r/R)), and operating with T R,r), as defined in Eq. fl), on the product function and after little algebraic simplification, one can obtain the following effective kinetic energy operator. [Pg.45]

Longuet-Higgins corrected the multivaluedness of the elechonic eigenfunctions by multiplying them with a phase factor, namely,... [Pg.82]

If now the nuclear coordinates are regarded as dynamical variables, rather than parameters, then in the vicinity of the intersection point, the energy eigenfunction, which is a combined electronic-nuclear wave function, will contain a superposition of the two adiabatic, superposition states, with nuclear... [Pg.106]

The electronic Hamiltonian and the comesponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on m. The index i in Eq. (9) can span both discrete and continuous values. The q ) form... [Pg.184]

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. [Pg.264]


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Adiabatic eigenfunction

Adjoint eigenfunctions

Angular momentum eigenfunction

Angular momentum eigenfunctions

Antisymmetric eigenfunctions of the spin

Bond eigenfunctions

Common set of eigenfunctions

Complete set of eigenfunctions

Completeness Schrodinger eigenfunctions

Coordinate operator, eigenfunction

Creeping flow eigenfunction expansions

Degenerate eigenfunctions

Determinantal eigenfunctions

Differentiability eigenfunction

Differentiability eigenfunctions

Diffusion eigenfunction

Dirac calculations eigenfunctions

Eigenfunction Hartree-Fock

Eigenfunction adjoint

Eigenfunction after

Eigenfunction cosine)

Eigenfunction expansion

Eigenfunction expansion application

Eigenfunction free particle

Eigenfunction g

Eigenfunction harmonic)

Eigenfunction nonstationary

Eigenfunction normalized

Eigenfunction of hermitian operators

Eigenfunction orbitals

Eigenfunction orthogonal

Eigenfunction orthonormality

Eigenfunction period

Eigenfunction principal

Eigenfunction prior

Eigenfunction reactivity

Eigenfunction spatial

Eigenfunction spin functions

Eigenfunction stationary

Eigenfunction, definition

Eigenfunction/eigenvalue

Eigenfunctions

Eigenfunctions Wigner transforms

Eigenfunctions and eigenvalues

Eigenfunctions antisymmetric

Eigenfunctions approximate

Eigenfunctions characterized

Eigenfunctions commuting operators have simultaneous

Eigenfunctions completeness

Eigenfunctions composition

Eigenfunctions description

Eigenfunctions dynamic correlations

Eigenfunctions effective Hamiltonians, mapping

Eigenfunctions equation

Eigenfunctions expansions

Eigenfunctions extracted from wavepacket dynamics Energy screening

Eigenfunctions for spin angular momentum

Eigenfunctions free particles

Eigenfunctions lowest-energy

Eigenfunctions mapping operators

Eigenfunctions model

Eigenfunctions nondegenerate

Eigenfunctions normalization

Eigenfunctions of Hamiltonian

Eigenfunctions of Hermitian operators

Eigenfunctions of angular momentum

Eigenfunctions of commuting operators

Eigenfunctions of the Smoluchowski equation

Eigenfunctions of the harmonic oscillator

Eigenfunctions orthogonality

Eigenfunctions parabolic coordinates

Eigenfunctions parent function

Eigenfunctions scalar product

Eigenfunctions separated wells

Eigenfunctions simultaneous

Eigenfunctions spherical coordinates

Eigenfunctions spheroconal coordinates

Eigenfunctions symmetry

Eigenfunctions table

Eigenfunctions unperturbed

Eigenfunctions well-behaved, defined

Eigenfunctions, Fock

Eigenvalue problems eigenfunctions

Eigenvalues eigenfunctions

Electromagnetic eigenfunctions

Electronic eigenfunction

Energy Levels and Eigenfunctions

Energy eigenfunction

Energy eigenfunctions

Energy eigenfunctions antisymmetric

Energy eigenfunctions electronic

Energy eigenfunctions hydrogenic

Energy eigenfunctions probability density

Energy eigenfunctions radial

Energy eigenfunctions radial factors

Energy eigenfunctions zero-order

Expansion in Terms of Eigenfunctions

Floquet eigenfunctions

General Eigenfunction Expansion

Greens Function by Eigenfunction (Mercers) Expansions

Hamiltonian eigenfunction

Hamiltonian eigenfunctions, calculation

Hamiltonian electronic, eigenfunctions

Hamiltonian operator eigenfunctions

Harmonic oscillator eigenfunction

Harmonic oscillator eigenfunctions

Harmonic oscillator energy eigenfunctions

Harmonic oscillator vibrational eigenfunctions

Hermitian operators eigenfunctions

Hydrogen atom energy eigenfunctions

Hydrogenic eigenfunctions

KS eigenfunctions

Kohn-Sham eigenfunctions

Linearly independent eigenfunctions

Lowest-energy eigenfunction

Many-particle operator eigenfunction

Matrix eigenfunctions

Methods for constructing spin eigenfunctions

Model eigenfunction

Molecular Rotation Eigenfunction

Momentum eigenfunctions

Nonhomogeneous Problem and Eigenfunction Expansion

Operator eigenfunctions

Orthogonal eigenfunctions

PT of the eigenfunctions and eigenvalues

Partial differential equation eigenfunctions

Position eigenfunctions

Product eigenfunction

Proof That Commuting Operators Have Simultaneous Eigenfunctions

Proof That Nondegenerate Eigenfunctions of a Hermitian Operator Form an Orthogonal Set

Radial eigenfunction

Radial eigenfunctions

Radial eigenfunctions table

Reactivity eigenfunctions

Resonance eigenfunction

Schrodinger eigenfunction

Schrodinger eigenfunctions

Spin Eigenfunctions

Spin angular momentum eigenfunctions

Spin eigenfunctions construction

Spin eigenfunctions symmetry properties

Spin-eigenfunction

Stationary eigenfunctions

Total Angular Momentum Eigenfunction

Total molecular eigenfunction

Transformed Hamiltonian eigenfunctions, calculation

Trial eigenfunctions

True eigenfunction

Two simple eigenfunctions of the spin

Variation method ground state eigenfunctions

Vibrating-rotator eigenfunctions

Vibrational eigenfunction

Vibrational eigenfunctions

Wavefunction eigenfunction

Zeroth-order eigenfunctions

Zeroth-order eigenfunctions expansion

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