Scalar product Dirac notation

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

For further convenience we also introduce a notation devised by Dirac and write the scalar product of ip and 0 as (0 ip), so that... 

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. 

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. 

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). 

Some notations. - Before starting the main part of this section, it may be convenient to discuss the use of some specific notations. Dirac [27] considered the bracket as the scalar product of a bra-vector , and - in this connection -he also introduced the ket-bra operators T = Ibxal defined through the relation Dc = b. They satisfy the relations T = T, Tt = laxbl, and trT = . 

The inner product (scalar product) (.,. ) in the Hilbert space Y follows from the canonical rules for inner products of direct sums and tensor products of Hilbert spaces like mentioned above. We use common Dirac notation for matrix elements (T O Z) of an operator O cuid vectors F) and Z) in the Hilbert space Y. 

Because Y is a potentially complex function including an imaginary part, Y designates the complex conjugate wave function. The compact and quite famous "bracket" notation on the right-hand side of Equation (2.7) bears the name of Dirac, and the "bra" (Y and "ket" Y) symbols stand for Y and Y and their integration. Mathematically, an integral such as / Y Ydr has been re-written as a scalar product (Y Y) within a complex vector space. 

In the first chapter, we saw that if we wanted to rotate the 2px function, we automatically also needed its companion 2py function. If this is extended to out-of-plane rotations, the 2/ function will also be needed. The set of the three p-orbitals forms a prime example of what is called a linear vector space. In general, this is a space that consists of components that can be combined linearly using real or complex numbers as coefficients. An n-dimensional linear vector space consists of a set of n vectors that are linearly independent. The components or basis vectors will be denoted as fi, with I ranging from 1 to n. At this point we shall introduce the Dirac notation [1] and rewrite these functions as / >, which characterizes them as so-called kef-functions. Whenever we have such a set of vectors, we can set up a complementary set of so-called fera-functions, denoted as /t I The scalar product of a bra and a ket yields a number. It is denoted as the bracket fk fi). In other words, when a bra collides with a ket on its right, it yields a scalar number. A bra-vector is completely defined when its scalar product with every ket-vector of the vector space is given. 

For the sake of completeness, we should note that Fe)mman introduced a compact notation for the Dirac equation by defining the scalar product of the four-component vector of all 7 matrices with any 4-vector (a ) = aP, a) by a slash through the symbol for this vector. 

This quantity, for which we employ the bra-ket ((, )) notation due to Dirac (1958), is called the Hermitian scalar product of the two functions 0, and orthonormal functions it is then easy to show that the best fit of the function / results when... 

Here and throughout the chapter, we use the bra and ket vector notation as introduced by P. Dirac to describe a specific state vector of a system. Whenever the scalar product of two state vectors, (i/rl and 1 ), is defined, then the complete bracket tf cp) = I p (j)cp(j)dr will denote a number. 

---