This chapter considers the idea of representing physical as well as nonphysical quantities by operators and collect together some frequently used operator techniques. It postulates that any quantity in quantum physics is specified by a linear operator. In a manipulation with operators, it is very important to maintain the order of the operators because operator algebra is not in general commutative for multiplication. An important concept in quantum physics is a commutator. The chapter focuses on to define a special class of operators called Hermitian operators. They are distinguished by the relationship of  and † The Hermitian operators are of great importance in quantum physics, as they represent physical (measurable) quantities. The chapter defines a very important property of wave functions: orthogonality. This property, for example, allows to identify whether the given wave functions belong to the same operator or not. The orthogonality condition of two functions is analogous to the orthogonality condition of two vectors.
This chapter considers the idea of representing physical as well as nonphysical quantities by operators and collect together some frequently used operator techniques. It postulates that any quantity in quantum physics is specified by a linear operator. In a manipulation with operators, it is very important to maintain the order of the operators because operator algebra is not in general commutative for multiplication. An important concept in quantum physics is a commutator. The chapter focuses on to define a special class of operators called Hermitian operators. They are distinguished by the relationship of  and † The Hermitian operators are of great importance in quantum physics, as they represent physical (measurable) quantities. The chapter defines a very important property of wave functions: orthogonality. This property, for example, allows to identify whether the given wave functions belong to the same operator or not. The orthogonality condition of two functions is analogous to the orthogonality condition of two vectors.
Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.
This chapter provides a simple general introduction to some basic mathematics necessary for a subsequent discussion of Lie algebras, operator methods, and their applications to theoretical physics and chemistry. It also provides some useful mathematical tools to derive important theoretical results and expressions in a simple and elegant way. In quantum mechanics the state of a system is represented by an element of a particular vector space, in which one defines an appropriate scalar or inner product, known in mathematics as Hilbert space. Hermitian operators with a complete set of eigenvectors belonging to that state space represent physical observables. Moreover, in certain cases it proves useful to work with vector spaces formed by linear operators. Hermitian operators are particular cases of linear operators and physical observables are particular cases of Hermitian operators. In many physical applications of operator methods one needs to reorder products of exponential operators or to rewrite one such operator as a product of simpler ones.
of theory What you need to know Definitions and properties Operator, linear operator, functions of operators, commuting and noncommuting operators, eigenfunction, eigenvalue, degeneracy, normalised function, orthogonal functions, Hermitian operator. Postulates of quantum mechanics (1) The state of a system with n position variables q 1 , q 2 , … q n is specified by a state (or wave) function ψ( q 1 , q 2 , … q n ). All possible information about the system can be derived from this state function. In general, n is three times the number of particles in the system. So for a single particle n = 3, and q 1 , q 2 , q 3 may be the Cartesian coordinates x, y, z , or the spherical polar coordinates r , θ, φ, or some other set of coordinates. (2) To every observable there corresponds a Hermitian operator given by the following rules: (i) The operator corresponding to the Cartesian position coordinate x is x × similarly for the coordinates y and z . (ii) The operator corresponding to p x , the x component of linear momentum, is (ħ/i)∂/∂ x – similarly for the y and z components. (iii) To obtain the operator corresponding to any other observable, first write down the classical expression for the observable in terms of x, y, z, p x , p y , p z , and then replace each of these quantities by its corresponding operator according to rules (i) and (ii). (3) The only possible result which can be obtained when a measurement is made of an observable whose operator is A is an eigenvalue of A .
Operators represent physically observable quantities, as discussed in Section 2.5. The structure and property of operators depend on the nature of the degree of freedom; operators act on the state space and in particular on the state vector of a given degree of freedom. The significance of operators in the interpretation of quantum mechanics has been discussed in Baaquie (2013e). The operators discussed in this chapter are mostly based on the continuous degree of freedom, which is analyzed in Section 3.1. Hermitian operators represent physically observable properties of a degree of freedom and their mathematical properties are defined in Section 3.3. The coordinate and momentum operators are the leading exemplar of a pair of noncommuting Hermitian operators and these are studied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, a finite-dimensional example of the shift and scaling operators; Section 3.8 provides a unitary representation of the coordinate and momentum operators. The term self-adjoint operator is used for Hermitian operators when there is a need to emphasize the importance of the domain of the Hilbert space on which the operators act – a topic not usually discussed in most books on quantum mechanics. Sections 3.10 and 3.11 discuss the concept of self-adjoint operators, in particular the crucial role played by the domain for realizing the property of self-adjointness. It is shown in Section 3.12 how the requirement of self-adjointness yields a non-trivial extension of Hamiltonians that include singular interactions.
A number of quantum states form an orthonormal and complete basis set if they are normalized and mutually orthogonal and any quantum state can be expressed as a linear combination. In classical mechanics, any two physical variables are commutable. However, this property is no longer valid for physical operators. Commutation of operators is an important property of quantum mechanics. Although there are many representations of wave functions and observables connected by unitary transformations, it is useful to distinguish certain classes of representations, called pictures, which differ in the way the time evolution of the system is treated. The Schrodinger picture is one in which the operators are time-independent but the wave function is time dependent. The virial theorem relates the kinetic energy to its potential in an energy eigenstate.
Linear algebra is the natural mathematical language of quantum mechanics. For this reason, the present chapter starts with a review of Hilbert spaces for vectors and operators. We introduce vector and Hilbert spaces along with inner products and metrics. The Dirac notation is developed for the Euclidean vector spaces as a starting point for the concepts of complete orthonormal sets of vectors, closure, dual vector spaces, and adjoint operators. The Dirac delta function in various forms and the principal part is introduced in Appendix 5 as an essential tool. The chapter turns to the main use of Dirac notation for function spaces; the concepts of norm, inner product, and closure are discussed. Fourier, Cosine, and Sine series are discussed as examples of expansions in complete orthonormal sets of functions. Although Hilbert spaces are interesting mathematical objects with important physical applications, the study of linear algebra remains incomplete without a study of linear operators (i.e., linear transformations). In fact, the set of linear transformations itself forms a vector space and therefore has a basis set. The basis set for the operator is linked with the basis sets for the spaces that it operates between. The linear operator can be discussed as an abstract operator or through an isomorphism as a matrix or as a generalized expansion in operator space. A Hermitian (a.k.a., self-adjoint) operator produces a basis set within a Hilbert space. The basis set comes from the eigenvector equation for the particular operator. The fact that a Hermitian operator produces a complete set (of orthornomal vectors) has special importance for quantum mechanics. Observables such as energy or momentum correspond to Hermitian operators. Complete sets make it possible to represent every possible result of a measurement of the observable by an object (vector) in the theory. The Hermitian operators have real eigenvalues which represent the results of the measurement.