The fundamental role of an operator is to effect a transformation. Differential operators and matrices are two important examples of how operators can be represented in a concrete way that makes quantitative calculations possible on systems that they represent. Both types of operator play important roles in quantum mechanics and the people shall meet them again. In a sense an operator interrogates a system represented by its eigenfunction, which results in a definite answer given by the eigenvalue. Operators with real eigenvalues also play a key role in quantum theory. Operators with real eigenvalues are conventionally represented by what are termed Hermitian or self-adjoint operators. Non-commutation has important consequences when it comes to information about two different parameters associated with a system. The connection between differentiation of a product, the Leibniz rule, and the commutator bracket, is indicative of the underlying algebraic structure of differential calculus.
The fundamental role of an operator is to effect a transformation. Differential operators and matrices are two important examples of how operators can be represented in a concrete way that makes quantitative calculations possible on systems that they represent. Both types of operator play important roles in quantum mechanics and the people shall meet them again. In a sense an operator interrogates a system represented by its eigenfunction, which results in a definite answer given by the eigenvalue. Operators with real eigenvalues also play a key role in quantum theory. Operators with real eigenvalues are conventionally represented by what are termed Hermitian or self-adjoint operators. Non-commutation has important consequences when it comes to information about two different parameters associated with a system. The connection between differentiation of a product, the Leibniz rule, and the commutator bracket, is indicative of the underlying algebraic structure of differential calculus.
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 basic quantity in quantum mechanics is the wave function ψ. Given a quantum mechanical system we write the Schrödinger equation for it. Its solution is the wave function. When introducing the wave function we stated that the maximum information about the state of a system can be obtained only from ψ. How do we extract observable properties of the system from the wave function? In quantum mechanics for every classical dynamical variable there exists an operator. For example, the operator of momentum p is −i~∇ while the operator of energy variable H = p2/(2m)+V (X) is −(~2/2m)∇2+V (X). Operation of an operator on ψ gives another function. If the new function is λψ, then λ is the eigenvalue of the operator. For example, the energy of a particle in a particular state represented by ψ can be obtained by operating H on ψ. That is, we have Hψ = Eψ where E is the energy eigenvalue. The expectation value of A(X,p) can be obtained from the formula 〈A〉 = ∫ ∞ −∞ ψ∗Aψ dτ , (3.1) where ψ is the solution of the equation Hψ = Eψ. Operators in quantum mechanics which represent observables are of a spe- cial type: They are linear and Hermitian. Such operators not only form computation tools, but also form the most effective language in terms of which the theory can be formulated. The fascinating point is that using linear operators we can understand the quantum mechanics. Therefore, it is important to study the basic theory of linear operators. In this chapter, we present important properties of linear operators and their eigenfunctions. We discuss the features of various operators which are necessary for studying quantum and ematical structure of quantum mechanics is.
Operators in quantum mechanics which represent observables are of a special type: They are linear and Hermitian. Such operators not only form computation tools, but also form the most effective language in terms of which the theory can be formulated. This chapter presents important properties of linear operators and their eigenfunctions. It discusses the features of various operators which are necessary for studying quantum mechanical systems. The chapter shows how rich and beautiful the mathematical structure of quantum mechanics is. There exists a class of eigenvalue problems, the so-called parity-time reversal (PT) -symmetric non-Hermitian Hamiltonian problems. They have arisen in recent years in a number of physical contexts such as localization transitions, vortex pinning and population biology. Such non-Hermitian Hamiltonians can be used to describe certain experimental phenomena. PT-symmetry has been noticed in certain optical systems, such as optical couplers, microwave billiard and large-scale temporal lattices.
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.
With the postulates already formulated in the language of an abstract inner product spaces, this chapter shows how an observable can be naturally expressed as a linear Hermitian operator acting on the inner product space where the states live. We start by introducing linear operators acting on vector spaces and their representations. Next, we discuss the systems of eigenvalues and eigenvectors associated with such operators. The important case of Hermitian operator and the properties of their eigenvalues and eigenvectors, relevant to quantum mechanics, are introduced. After developing the necessary background in linear algebra, the connection of Hermitian operators with quantum observables is established. We then proceed to explain what this connection actually accomplishes. In particular, we show how it translates the target questions of quantum mechanics into an eigenvalue problem. This makes it evident what the nature of the input to a quantum theory should be. This input enters the formalism through what are known as “quantization rules”. I discuss this in the last section of this chapter. An illustration using a disguised spin-half system winds up the chapter.
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 .