Linear Operators and Transformation Matrices

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. 

Ceulemans, Group Theory Applied to Chemistry. Theoretical Chemistry and Computational ModeUing, DOl 10.1007/978-94-007-6863-5 2, 

This result can be summarized with the help of the Kronecker delta, 8ij, which is zero unless the subscript indices are identical, in which case it is unity. Hence, for an orthonormal basis set. 

In quantum mechanics, the bra-function of / is simply the complex-conjugate function, fk, and the bracket or scalar product is defined as the integral of the product of the functions over space  

A linear operator is an operator that commutes with multiplicative scalars and is distributive with respect to summation this means that when it acts on a sum of functions, it will operate on each term of the sum  

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