Dirac notation

The spectral theorem can also be used to express many functions of A, by recognizing that all powers of A have the same eigenvectors as A and the associated eigenvalues are equivalent functions of the a . 

The powerful concepts of matrix algebra can also be further extended to partitioned matrices, whose elements are themselves matrices rather than scalars [cf. (9.7)]  

So long as dimensional conformability is maintained, such super-matrices (matrices of matrices) obey additive, scalar-multiplicative, and matrix-multiplicative equations analogous to (9.8)—(9.11), such as 

The matrix-algebraic representation (9.20a-e) of Euclidean geometrical relationships has both conceptual and notational drawbacks. On the conceptual side, the introduction of an arbitrary Cartesian axis system (or alternatively, of an arbitrarily chosen set of basis vectors ) to provide vector representations v of geometric points V seems to detract from the intrinsic geometrical properties of the points themselves. On the notational side, typographical resources are strained by the need to carefully distinguish various types of 

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

---

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

Since the Dirac notation suppresses the variables involved in the integration, we re-express the orthogonality relation in integral notation... 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

The functions tpi(x) are, in general, complex functions. As a consequence, ket space is a complex vector space, making it mathematically necessary to introduce a corresponding set of vectors which are the adjoints of the ket vectors. The adjoint (sometimes also called the complex conjugate transpose) of a complex vector is the generalization of the complex conjugate of a complex number. In Dirac notation these adjoint vectors are called bra vectors or bras and are denoted by or (/. Thus, the bra (0,j is the adjoint of the ket, ) and, conversely, the ket j, ) is the adjoint (0,j of the bra (0,j... 

The bracket (bra-c-ket) in

) provides the names for the component vectors. This notation was introduced in Section 3.2 as a shorthand for the scalar product integral. The scalar product of a ket tp) with its corresponding bra (-01 gives a real, positive number and is the analog of multiplying a complex number by its complex conjugate. The scalar product of a bra tpj and the ket Aj>i) is expressed in Dirac notation as (0yjA 0,) or as J A i). These scalar products are also known as the matrix elements of A and are sometimes denoted by Ay. 

Combining these two equations gives equation (3.29), which when expressed in Dirac notation is... 

If the eigenvalues of N are represented by the parameter X and the corresponding orthonormal eigenfunctions by (pxi( ) or, using Dirac notation, by Xi), then we have... 

In the case of a closed shell molecule, using Dirac notation the expression becomes... 

The expansion coefficients on the right hand side of Eq. (1.25) are the Clebsch-Gordan coefficients.2 The eigenfunctions of the angular momentum, which can be written, abstractly, using Dirac notation 11, m >, satisfy the equations (h=1)... 

Let s assume a wave function of the Slater determinant form and find an expression for the expectation value of the energy. We ve written a Slater determinant as a ket vector in shorthand notation, allowing us to make use of Dirac notation for such things as overlap. In this context, recall that... 

Yes, I know. Very confusing. But it s all just notation, and can be understood. In physicist s notation (equivalent to Dirac notation), tpitpjitpk tpi) refers to the two electron integral where and are functions of electron 1, while -j and ipi are functions of electron 2. Chemist s notation (with the square brackets []) places the functions of electron 1 on the left and the functions of... 

Within the old adiabatic approximation, Eq. (39) is the basic starting point. However, from here on, the various approximations diverge. For ease of discussion, we shall first still make the Condon approximation, and then give the further approximations. However, it must be kept in mind that many similar approximations are also made in papers that use a non-Condon approach. The basic premise of the Condon approximation is that the electronic part of the matrix element varies sufficiently slowly with Q so that it can be taken out of the integration over dQ. The matrix element then reduces to products of electronic and vibrational integrals. In Dirac notation... 

It is important to recognize that the small subset of matrix equations introduced in the main text (typically, restricted to real matrix elements) will be found sufficient to exploit the geometrical simplicity that underlies equilibrium thermodynamics. Nevertheless, it is useful to introduce the thermodynamic vector geometry in the broader framework of matrix theory and Dirac notation that is broadly applicable to the advanced thermodynamic topics of Chapters 11-13, as well as to many other areas of modem physical chemistry research. 

To further illustrate Dirac notation for some simple formulas in Euclidean 3-space, we can rewrite analogs of (9.20a-e) in Dirac notation, all in terms of underlying Dirac objects y) (using boldface symbols to stress the association with ordinary vectors) ... 

Use of Dirac notation allows us to recognize at a glance that v) is a column vector, (v is the adjoint row vector, (v v) is the scalar product of these two vectors, and v)(v is a corresponding matrix dyadic, all referring to underlying object v. Further examples of Dirac notation are shown in Sidebar 9.2. 

The basic concept of Dirac notation is that a matrix A is only a representation of an underlying operator A in a chosen set of basis vectors u -) (written as kets)... 

Problem Write the eigenvalue problem (S9.1-15) in Dirac notation, and prove the fundamental properties (S9.1-16), (S9.1-17) of the eigenvectors and eigenvalues for Hermitian A, assuming the eigenvalues are nondegenerate (unequal). 

It will be useful practice for the physical chemistry student to rewrite other matrix and vector equations of Sidebar 9.1 in Dirac notation, both for future applications to quantum theory as well as the intended present application to equilibrium thermodynamics. 

Euclidean geometry was originally deduced from Euclid s five axioms. However, it is now known that necessary and sufficient criteria for Euclidean spatial structure can be stated succinctly in terms of distances, angles, and triangles, or, alternatively, the scalar product of the space. We can express these criteria by employing Dirac notation for abstract ket vectors R ) of a given space M with scalar product (R R7). 

---