Matrix representation of an operator

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

For a matrix representation of an operator. A, the projection onto the x subspace is given by pre- and post-multiplying with a Q matrix defined as the outer vector product of xt, or the function equivalent in a ket-bra notation. 

Let us now consider how the matrix representations of an operator (P are related in two different complete orthonormal bases. The result we shall obtain plays a central role in the next subsection where we consider the eigenvalue problem. Suppose O is the matrix representation of ( ) in the basis i>, while il is its matrix representation in the basis a> ... 

Exercise 1.15 As a further illustration of the consistency of our notation, consider the matrix representation of an operator 0 in the basis f(x). Starting with... 

This form of the matrix equation displays the structure of the matrix representation of an operator that is symmetric under time reversal, given in (10.30)... 

H is called the matrix representation of the Hamiltonian operator. A matrix representation of an operator is a matrix of integral values arranged in rows and columns according to the basis functions. Clearly, the values in a matrix representation are dependent on the functions that were selected for the basis set. A different basis set implies a different matrix representation. Wherever it is important to keep track of the basis used in the representation, a superscript is added to the designation of the matrix, for example, W, and it identifies the particular function set. Now, the quantity in Equation C.19 can be written with the coefficient vectors and the Hamiltonian matrix in a very simple form. 

If a proper rotation is combined with a reflection with respect to the axis of rotation, it is called an improper rotation The matrix representation of silvan operation is found simply by replacing 1 by -1 in Eq. (104), The Scftfllfllies symbol for an improper rotation by y is S /tp- Hence, matrix the representation of an improper counter-clockwise rotation by y is of the form ... 

A.10-3. Proof that the matrix olomonts of an operator H which commiiteo with all Og of a group vanish between functions belonging to different irreducible representations... 

From an all-electron calculation we have available the matrix representations of the operators /c and Rc in the all-electron basis. Keeping the same basis we have sought coefficients At) for each / value, such that... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

Before we go further, we must keep in mind that the matrix elements of an operator remains unmodified by changing the representation. This may be verified by inserting a unity operator at the right and at the left of the operator B//, to give... 

Eq. (36) is a version of the Heisenberg representation of an operator in the L-space, which will be useful in later manipulations. Similarly the time correlation functions between two operators given in Eq. (26) in H-space may also be re-expressed as a matrix element of the above form in the L-space. Thus in the... 

Every primitive ket is an eigenfunction of an operator of this form and the matrix representation of this operator in the ket space is diagonal, i.e. m ala, k = If this operator is summed over all spin orbitals, the... 

All of the above hinges on the assumption that /i is a representation of an operator h which is invariant with respect to the operations of the molecular point group. Obviously the one-electron Hamiltonian and the unit operator satisfy this condition and so the matrices h and S of the LCAO method can be symmetry blocked in this way. We have seen that, in general, the matrix representation of the Hartree-Fock operator will not satisfy this condition, so that it is a constraint on the LCAO method to make the assumption that the matrix can be treated in this way. As we have seen, this constraint consists of generating self-consistent symmetries which in certain critical cases may prevent us from obtaining the lowest-energy determinant. 

The above graphical constructs represent individual normal-ordered operators. Using these graphical representations one can derive all unique combinations (operator contractions) which contribute to the considered matrix element of an operator product. The rules for the diagram manipulations are standard and can be found elsewhere [13,51]. General-order coupled cluster equations can be derived from the general-order coupled cluster functional ... 

We say that O is the matrix representation of the operator 0 in the basis ej. The matrix O completely specifies how the operator 6 acts on an arbitrary vector since this vector can be expressed as a linear combination of the basis vectors and we know what ( ) does to each of these basis vectors. 

This last equation is the basis for an iterative numerical scheme for a chosen basis set expansion of the two-component spinor with resulting matrix representation of all operators [616]. For this purpose the equation has first to be multiplied by the operator F rr-Pj, from the left in order to reduce it to a computationally feasible form, where... 

Although Eq. (11.103) seems to be unnecessarily complicated, it can be solved by purely numerical iterative techniques, and the matrix representation of the operator Q is obtained [616]. This result appears to be the best representation of the operator Q that can be achieved within a given basis and is only limited by machine accuracy. Note that all expressions occurring in Eq. (11.103) and hence the matrix representation of the operator Q depend only on the squared momentum rather than on the momentum variable itself, which is the key feature of the computational feasibility of this approach. This is an essential trick for actual calculations of transformed two-component operators first noticed by Hess [623] (compare also section 12.5.1). Eq. (11.103) is still nonlinear and therefore bears the possibility of negative-energy solutions for the operator Q. The choice toward the positive-energy branch has to be implemented via the boundary conditions imposed on the numerical iterative technique. Essentially, Q and hence R have to be small operators with operator norms much smaller than unity. 

It has been noted (footnote on p.l24) that p(xi xl) formally resembles a matrix element, in which Xi andxj play the part of (continuous) row and column indices in this sense it provides a particular representation of p. We now note that, on introducing any orthonormal set i/ r(JCi) > the array of coefficients appearing in (6.4.1) simply provides a true matrix representation of the operator p, in which Xj, x[ are replaced by the discrete indices r, s. This follows easily from the definition of the matrix elements of an operator since, using the orthonormality property,... 

A number of procedures have been proposed to map a wave function onto a function that has the form of a phase-space distribution. Of these, the oldest and best known is the Wigner function [137,138]. (See [139] for an exposition using Louiville space.) For a review of this, and other distributions, see [140]. The quantum mechanical density matrix is a matrix representation of the density operator... 

In an n-dimensional space L, the linear operators of the representation can be described by their matrix representatives. This procedure produces a homomorphic mapping of the group G on a group of n x n matrices D(G), i.e., a matrix representation of the group G. From equations (6) it follows that the matrices are non-singular, and that... 

The matrix representation of the spin operator requires the spin state of a particle to be represented by row vectors, commonly interpreted as spin up or down. An arbitrary state function J must be represented as a superposition of spin up and spin down states... 

Each of these columns of this symmetrical matrix may be seen as representing a molecule in the subspace formed by the density functions of the N molecules that constitute the set. Such a vector may also be seen as a molecular descriptor, where the infinite dimensionality of the electron density has been reduced to just N scalars that are real and positive definite. Furthermore, once chosen a certain operator in the MQSM, the descriptor is unbiased. A different way of looking at Z is to consider it as an iV-dimensional representation of the operator within a set of density functions. Every molecule then corresponds to a point in this /V-dimensional space. For the collection of all points, one can construct the so-called point clouds, which allow one to graphically represent the similarity between molecules and to investigate possible relations between molecules and their properties [23-28]. 

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