Matrix representatives of operators

In eqs. (7a) and (7b) v) is a matrix of one column containing the components of v, and u is a matrix of one row, which is the transpose of w ), the matrix of one column containing the components of u, complex conjugated. In eq. (6), transposition is necessary to conform with the matrix representation of the scalar product so that the row x column law of matrix multiplication may be applied. Complex conjugation is necessary to ensure that the length of a vector v 

Suppose a basis (e is transformed into a new basis (e under the proper rotation R, so that 

To find the ith component v take the scalar product of e, with v. Here the basis is real and orthonormal, so 

We now represent the transformed basis vectors e/ in terms of the original set e, by expressing each as the sum of its projections, according to eq. (3). Writing each e/ (j= 1, 2, 3) as the sum of their projections along e, yields 

Example 3.2-3 Find the t ransformed components of a vector r when acted on by the operator C4z = R n/2 z). 

---

The operator q2 is the square of the operator q. Since matrix representatives of operators obey the same relations as the operators (Section 2.3), the q2 matrix [whose elements are ] is the square of the q matrix [whose elements are (4.45)]. Hence, using the matrix-multiplication rule (2.11), we have... 

Matrices are rectangular arrays of numbers and obey the rules (7.105)-(7.107) for addition, scalar multiplication, and matrix multiplication. If / is a complete, orthonormal set of functions, then the matrix A with elements A = fm A fn) represents the linear operator A in the / basis. Also, the column matrix u with elements m, equal to the coefficients in the expansion u = represents the function u in the / basis. The matrix representatives of operators and functions in a given basis obey the same relations as the operators and functions. For example, if C = A + B,R = Sf, and w = Au, then C = A -I- B, R = ST, and w = Au. 

The functions (2.50) are called basis functions The matrices F, G,. .. are called matrix representatives of the operators F, G,. .. in the

specific form of the matrix representation of a set of operators depends on the basis chosen. Equation (2.53) shows that the effect of the operator G on the basis functions is determined by the matrix elements GkJ. Since an arbitrary well-behaved function can be expanded using the complete set (2.50), knowledge of the matrix G allows one to determine the effect of the operator G on an arbitrary function. Thus, knowledge of the square matrix G is fully equivalent to knowledge of the corresponding operator G. Since G is a Hermitian operator, its matrix elements satisfy Gij = (GJi). Hence the matrix G representing G is a Hermitian matrix (Section 2.1). 

U m T(R) a unitary operator time-evolution operator (Section 13.1) matrix representative of the symmetry operator R sometimes just R, for brevity... 

The worksheet displayed by the GT Calculator files by the action of the Reduce a Character command button, Figure 1.5, is shown in Figure 1.12. This worksheet takes as input the character, F, normally a reducible representation, i.e. a set of traces of the matrix representatives of the operators of the group and returns the direct sum components of this character, identified by Mulliken symbols. This input is entered in the red-bordered cells and the direct sum components are returned as numbers of Mulliken symbols in the last row of the display. 

Usually, in chemistry texts, the individual character values are identified as x ( ) for the character under operation R and the x ( ) are the traces [the sums of the diagonal matrix elements of the matrix representatives of the symmetry operations. 

We must diagonalize the matrix representative of the excitation operator Hq—Eq over the appropriate basis. 

These B m can be considered as the elements of a matrix, representing the operator B in the system of the proper functions, of the... 

When the direct product of two irreducible matrix representations of a group is reducible, it can be reduced to a direct sum of irreducible representations by cin equivalent transformation with a constant matrix, i.e. the same matrix for all the matrix representatives of the symmetry operators of the group (2). We shall assume the irreducible representations in unitary form then the constant matrix can be chosen as the real orthogonal matrix whose elements are the coupling coefficients occuring in Eq. (5). The orthogonality properties can be expressed as... 

A rotation about an axis transforms a left hand into a left hand and a right hand into a right hand. It conserves the chirality. It is called an operation of the first type. The determinant of any matrix representing an operation of the first typeis AJ= +1. 

We have proved that the matrix representatives of linear operators in a complete orthonormal basis set obey the same equations that the operators obey. 

We have used the complete, orthonormal basis / to represent the operator A by the matrix A of (7.107). Tlie basis / can also be used to represent an arbitrary function u, as follows. We expand u in terms of the complete set /j, according to M = 2i Ujfj, where the expansion coefGdents m, are numbers (not functions) given by Eq. (7.40) as , = fi u). Ihe set of expansion coefficients Hj, 2,... is formed into a column matrix (column vector), which we call u, and u is said to be the representative of the function u in the /j basis. If Au = w, where w is another function, then we can show (Problem 7.49) that Au = w, where A, u, and w are the matrix representatives of A, u, and w in the / basis. Thus, the effect of the linear operator A on an arbitrary function u can be found if the matrix representative A of A is known. Hence, knowing the matrix representative A is equivalent to knowing what the operator A is. 

Find the matrix representative of the unit operator 1 in a complete, orthonormal... 

Matrix representatives of the spin operators are considered in Problem 10.21. 

The magnetization of the S multiplet is calculated as -dE/dB by diagonalizing the matrix representative of H plus the Zeeman operator. It must be stressed, however, that the determination of the sign of the zfs parameter D, which gives information on the type of magnetic anisotropy, is extremely difficult on the basis of polycrystalline powder measurements only, unless the low temperature limit, where k T is small compared to D is reached. 

We have proved that the matrix representatives of linear operators in a complete orthonormal basis set obey the same equations that the operators obey. Combining Eqs. (7.109) and (7.110), we have the useful sum rule... 

The character we assign for a particular basis vector has been linked to the diagonal element in the operation s matrix. This can be generalized to say that the sum of the diagonal elements of a matrix representing an operation on a particular basis is the sum of the characters for the basis under that operation. In matrix algebra, the sum of the diagonal elements of a matrix A is known as the trace of the matrix, Tr(A), i.e. 

This matrix is a matrix representative of the 1-electron density operator p, as mentioned in Section 5.3 in the basis the operator is... 

Again, it is easily verified that J(p) and K(p) are matrix representatives of the operators introduced in (6.1.10) whose effect on a one-electron function is defined through (6.1.11). 

---