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Eigenvalues

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... [Pg.888]

In our experiment we used thresholing value 6 = 0.025, yielding m = 25 and a ratio between tbe smallest and the largest eigenvalue 4 = 0.01. [Pg.890]

While not unique, the Scluodinger picture of quantum mechanics is the most familiar to chemists principally because it has proven to be the simplest to use in practical calculations. Hence, the remainder of this section will focus on the Schrodinger fomuilation and its associated wavefiinctions, operators and eigenvalues. Moreover, effects associated with the special theory of relativity (which include spin) will be ignored in this subsection. Treatments of alternative fomuilations of quantum mechanics and discussions of relativistic effects can be found in the reading list that accompanies this chapter. [Pg.5]

To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... [Pg.7]

If the system property is measured, the only values that can possibly be observed are those that correspond to eigenvalues of the quantum-mechanical operator 4. [Pg.8]

An illustrative example is provided by investigating the possible momenta for a single particle travelling in the v-direction, p First, one writes the equation that defines the eigenvalue condition... [Pg.8]

The probability of measuring A = where is the eigenvalue associated with the normalized eigenfiinction ( ), is precisely equal to r> - For degenerate eigenvalues, the probability... [Pg.10]

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]

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]

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). [Pg.14]

The Flamiltonian commutes widi the angular momentum operator as well as that for the square of the angular momentum I . The wavefiinctions above are also eigenfiinctions of these operators, with eigenvalues tndi li-zland It should be emphasized that the total angular momentum is L = //(/ + )/j,... [Pg.23]

The simplest way to obtain X is to diagonalize S, take the reciprocal square roots of the eigenvalues and then transfomi the matrix back to its original representation, i.e. [Pg.39]


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A 2 x 2 generalized eigenvalue problem

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Adjacency Matrices and Their Eigenvalues for Toroidal Polyhexes

An Eigenvalue Approach to Coupled Cluster Theory

Analysis using eigenvalues

Angular momentum eigenvalues

Application of Deans Negative Eigenvalue Theorem to Aperiodic Polymers

Applying eigenvalue analysis to experimental design

Atomic Eigenvalues and Electronic Configurations of the Atom

Atomic eigenvalue problem

Behaviour of eigenvalues at Hopf bifurcation

Bifurcation from a Simple Eigenvalue

Bifurcation zero-eigenvalue

Bound motion eigenvalues

Bounds on the burning-rate eigenvalue

Burning rate - an eigenvalue

Burning-rate eigenvalue

Burning-rate eigenvalue bounds

Casimir operators eigenvalues

Closed-loop eigenvalues

Common Eigenvalues

Complex eigenvalue Schrodinger equation

Complex eigenvalue Schrodinger equation CESE)

Complex eigenvalues

Configurational eigenvalues

Conformation eigenvalues

Conjugate eigenvalue problem

Continuous eigenvalues

Continuous spectrum, of eigenvalues

Converged eigenvalues

Cycle-State Structure from Global Eigenvalue Spectrum

DAEs and the Generalized Eigenvalue Problem

Density functional eigenvalues

Desired eigenvalues

Determining eigenvalues and eigenvectors

Diagonal matrix of eigenvalues

Differentiability eigenvalue

Differentiability eigenvalue equation

Differential eigenvalues

Dirac equation eigenvalue spectrum

Dirac-Fock eigenvalue

Direct eigenvalue problem

Discrete eigenvalues

Discrete spectrum, of eigenvalues

Distribution of eigenvalues

Dominant eigenvalue

Dominant-eigenvalue method

Eigenfunction/eigenvalue

Eigenfunctions and eigenvalues

Eigenvalue 2-positivity conditions

Eigenvalue Based Methods

Eigenvalue Graetz problem

Eigenvalue Moment Method

Eigenvalue Monte Carlo techniques

Eigenvalue Tables

Eigenvalue after

Eigenvalue algorithms

Eigenvalue analyses eigenvalues

Eigenvalue analysis

Eigenvalue analysis Singular Value Decomposition

Eigenvalue analysis characteristic value

Eigenvalue analysis condition number

Eigenvalue analysis determinant

Eigenvalue analysis dynamic stability

Eigenvalue analysis dynamic system

Eigenvalue analysis eigenvector

Eigenvalue analysis multiplicity

Eigenvalue analysis numerical calculation

Eigenvalue analysis orthogonal matrix

Eigenvalue analysis quantum mechanics

Eigenvalue analysis similar matrices

Eigenvalue analysis systems

Eigenvalue characterized

Eigenvalue conduction

Eigenvalue decomposition

Eigenvalue descriptor

Eigenvalue dynamic correlations

Eigenvalue dynamical matrix

Eigenvalue equation condition

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Eigenvalue equation integral form

Eigenvalue equation leaky modes

Eigenvalue equation level systems

Eigenvalue equation local modes

Eigenvalue equation matrix form

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Eigenvalue equation structure

Eigenvalue equations

Eigenvalue equations Hamiltonian diagonalization

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Eigenvalue equations approximation

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Eigenvalue equations systems

Eigenvalue equations, reduced

Eigenvalue filtering

Eigenvalue following method

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Eigenvalue index

Eigenvalue lattice

Eigenvalue natural orbital functionals

Eigenvalue of a matrix

Eigenvalue of an operator

Eigenvalue positive dominant

Eigenvalue positive, simple

Eigenvalue prior

Eigenvalue problem, Sturm-Liouville

Eigenvalue problem, reduced

Eigenvalue problems eigenfunctions

Eigenvalue problems eigenvalues

Eigenvalue problems in quantum mechanics

Eigenvalue problems, solution

Eigenvalue reduced Hamiltonians

Eigenvalue relation

Eigenvalue second-order correction

Eigenvalue smallest

Eigenvalue spacing

Eigenvalue spectrum

Eigenvalue spectrum Dirac

Eigenvalue spectrum pseudopotential

Eigenvalue theory

Eigenvalue time-independent--------problem

Eigenvalue total angular momentum

Eigenvalue zero-field

Eigenvalue, Hartree-Fock molecular

Eigenvalue, vibrational

Eigenvalue-eigenvector method

Eigenvalue-following

Eigenvalue/eigenvector problem

Eigenvalue/eigenvector problem generalized matrix

Eigenvalues Dirac free-particle

Eigenvalues Floquet Hamiltonian

Eigenvalues Schrodinger equation

Eigenvalues Subject

Eigenvalues and Eigenvectors of the Matrix

Eigenvalues and Eigenvectors of the Rouse-Mooney Matrix

Eigenvalues and Eigenvectors, Diagonalizable Matrices

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors defined

Eigenvalues and eigenvectors of a symmetric matrix

Eigenvalues and positions of equilibrium points

Eigenvalues angle

Eigenvalues approximate

Eigenvalues atomic spinor

Eigenvalues auxiliary system

Eigenvalues calculations using Slater orbitals

Eigenvalues calculations using canonical

Eigenvalues calculations with Gaussian functions

Eigenvalues completeness

Eigenvalues definition

Eigenvalues degenerate

Eigenvalues description

Eigenvalues diffusivity matrix

Eigenvalues dressed Hamiltonian

Eigenvalues eigenfunctions

Eigenvalues equal

Eigenvalues for orbital angular momentum

Eigenvalues for spin angular momentum

Eigenvalues functions

Eigenvalues geometry

Eigenvalues hydrogen bonds

Eigenvalues hypersurfaces

Eigenvalues introduced

Eigenvalues many-body equations

Eigenvalues negative

Eigenvalues nondegenerate

Eigenvalues occurring from approximations

Eigenvalues of A and

Eigenvalues of Casimir operators

Eigenvalues of Hermitian operators

Eigenvalues of lx and ly

Eigenvalues of matrix

Eigenvalues of the Casimir operators

Eigenvalues orbitals-molecular

Eigenvalues orbitals-self-consistent field

Eigenvalues orthonormality

Eigenvalues parabolic coordinates

Eigenvalues persistent real

Eigenvalues perturbed

Eigenvalues potential

Eigenvalues potential function

Eigenvalues proof

Eigenvalues quantum dynamics

Eigenvalues rotational motions

Eigenvalues separated wells

Eigenvalues spherical coordinates

Eigenvalues transformations

Eigenvalues, continuous distribution

Eigenvalues, critical case

Eigenvalues, eigenvectors and band

Eigenvalues, ground-state valence level

Eigenvalues, principal component

Eigenvalues, principal component analysis

Energy eigenvalues

Energy eigenvalues states

Energy eigenvalues zero-order

Energy eigenvalues, orbital Schrodinger equation

Estimating eigenvalues Gershgorins theorem

Exchange potential eigenvalues

Finding Eigenvalues by Diagonalization

First Eigenvalues by Eq

First order extended eigenvalue problem

Floquet eigenvalue

Fock eigenvalues

General properties of the eigenvalues

Generalized Matrix Eigenvalue Equation

Generalized eigenvalue decomposition

Generalized eigenvalue problem

Generalized matrix eigenvalue problem

Groups, continuous eigenvalues

Hamiltonian eigenvalue

Hamiltonian energy eigenvalues

Harmonic oscillator eigenvalues

Hartree Fock eigenvalue functions

Hartree-Fock approximation energy eigenvalue

Hartree-Fock method eigenvalues

Hartree-Fock molecular orbital eigenvalue

Hermite equation, eigenvalue

Hermitian operator eigenvalues

Hessian eigenvalues

Hessian matrix eigenvalues

Higher order eigenvalues

Highest-occupied molecular orbital energy eigenvalue

Hilbert space eigenvalue equation

Huckel Molecular Orbital Theory 1 Eigenvalues

Hydrogen atom energy eigenvalues

Hydrogen energy eigenvalues

INDEX Eigenvalue problem

Identity operator eigenvalue

Interpretation of the LCAO-MO-SCF Eigenvalues

Invariant eigenvalue

Inverse eigenvalue problem

Jacobi eigenvalue problem

Jacobi eigenvalues

Kohn-Sham eigenvalues

Kohn-Sham orbital eigenvalues

Lame functions eigenvalue

Largest eigenvalue

Leading eigenvalues

Linear expansions and eigenvalue equations

Liouville eigenvalues

Liouville operator, eigenvalues

Liouvillian eigenvalue problem

Liouvillian eigenvalues

Lost eigenvalues

MATLAB eigenvalues

Matrices density matrix eigenvalues

Matrices eigenvalues/eigenvectors

Matrices, Eigenvalues, and Eigenvectors

Matrix Methods for the One-Dimensional Eigenvalue Schrodinger Equation

Matrix diagonalization eigenvalues

Matrix eigenvalue analysis

Matrix eigenvalue equation

Matrix eigenvalue problem

Matrix eigenvalues

Matrix representation of the noninteracting eigenvalue problem

Model Updating Using Eigenvalue-Eigenvector Measurements

Model eigenvalue problem

Molecular eigenvalue equation

Molecular eigenvalues

Molecular orbital theory eigenvalue equation

Natural orbital eigenvalue equation

Near zero eigenvalues

Negative-eigenvalue theorem

New eigenvalues

Non-degenerate eigenvalues

Nonhermitian eigenvalue problem

Normal matrix eigenvalue properties

Numerical calculation of eigenvalues and eigenvectors in MATLAB

Observables eigenvalue equation

On the Behaviour of Eigenvalues in Adiabatic Processes

One-electron eigenvalue

Open-loop eigenvalues

Operator eigenvalues

Orbital angular momentum eigenvalues

PT of the eigenfunctions and eigenvalues

Partial differential equation eigenvalues

Particle-hole eigenvalue

Persistent eigenvalues

Problem eigenvalue

Proof That Eigenvalues of Hermitian Operators Are Real

Properties of Eigenvalues

Properties of Eigenvalues and Eigenvectors

Pseudo-eigenvalue problem

Quadratic eigenvalues

Radial Eigenvalue Equations

Rank and Eigenvalues

Reduced eigenvalue

Relative motion, energy eigenvalue

Response function eigenvalue problem

Rovibrational eigenvalues

Schrodinger eigenvalues

Secular Equations and Eigenvalues

Semiclassical resonance eigenvalues

Simple eigenvalue

Simple leading eigenvalues

Single-Particle Eigenvalues and Excited-State Energies

Singular eigenvalues

Solution at two equal eigenvalues

Solution for complex eigenvalues

Solution of the Energy Eigenvalue Problem

Spin angular momentum eigenvalues

Spin eigenvalues

Spin eigenvalues orbit coupling

Spin eigenvalues relaxation

Spurious eigenvalues

Stability eigenvalue analysis

Stability eigenvalues

Stress tensor eigenvalue

Sturm-Liouville eigenvalue

Superoperator eigenvalue problem

Tensor eigenvalue

The EOM-CC eigenvalue problem

The Eigenvalue Method

The Eigenvalue Problem

The Postulate Relating Measured Values to Eigenvalues

The QR method for computing all eigenvalues

The eigenvalue equation

The eigenvalue spectrum

The matrix eigenvalue equation

True Zero Eigenvalues Catastrophe Points

Two Dimensional Eigenvalue Schrodinger Equation

Unperturbed energy eigenvalue

Vibrational eigenvalue problem

Zero Eigenvalues of the Hessian

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