According to quantum mechanics, the information with respect to any measurement on a physical system is contained in a mathematical object, the wave function. In this chapter we become familiar with the mathematical objects that represent the measured properties themselves, namely the quantum mechanical operators. We start from a brief introduction into operators and their properties, emphasizing linear operators, and noncommuting operators. Then we introduce the canonical position and momentum operators. Defining functions of operators, we derive different quantum mechanical operators that correspond to different physical observables, including angular momentum, kinetic energy, and the scalar potential energy. Finally, we introduce the quantum mechanical total energy operator (the Hamiltonian) and demonstrate its explicit generic form for nanoscale building blocks such as atoms and molecules.
According to quantum mechanics, the information with respect to any measurement on a physical system is contained in a mathematical object, the wave function. In this chapter we become familiar with the mathematical objects that represent the measured properties themselves, namely the quantum mechanical operators. We start from a brief introduction into operators and their properties, emphasizing linear operators, and noncommuting operators. Then we introduce the canonical position and momentum operators. Defining functions of operators, we derive different quantum mechanical operators that correspond to different physical observables, including angular momentum, kinetic energy, and the scalar potential energy. Finally, we introduce the quantum mechanical total energy operator (the Hamiltonian) and demonstrate its explicit generic form for nanoscale building blocks such as atoms and molecules.
An overview of main operators in quantum mechanics is presented. Operators for position, linear momentum, kinetic energy, potential energy, total energy, and angular momentum are introduced. Properties of linear operators are described. Commutative property of operators has profound implications because a pair of non-commutative operators cannot be measured accurately at the same time. Eigenfunctions and eigenvalues of an operator are defined, and operator form of Schrodinger’s equation is given. Most operators in quantum mechanics are of the Hermitian category because they always have real eigenvalues. The condition to test Hermitian nature of operators is specified. Postulates of quantum mechanics are stated to serve as a convenient starting point.
Abstract According to quantum mechanics, all the dynamical information about a system is contained in its °wave function. The outcome of an observation of a particular property can be predicted by performing the appropriate mathematical operation on the wave function. Each physical observable is therefore represented by an operator, a mathematical action such as multiplication or differentiation. The operator corresponding to the observable Q is often denoted fi, but we shall not make that distinction here. Different choices of operators for observables correspond to the different representations of quantum mechanics.
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.
Several somewhat independent concepts are introduced in this chapter. First, we've stated that the wave equation contains information about the particle. How is the information extracted? By using operators, which often include derivative operations. By "operating" on the wave function, we can pull out certain factors, such as the momentum or the energy of the particle. Here, three operators are discussed: the total energy operator in terms of a time-derivative; a more space-based energy operator called the Hamiltonian, which simply represents the kinetic energy plus the potential energy; and the momentum operator. Second, we show that an oscillating wave function does not necessarily mean that the particle position is oscillating, since the position is related to the square modulus of the wave function. Third, the Heisenberg uncertainty principle is introduced using a free particle with specified energy, but with no specified position. Fourth, the idea of "quantization" is introduced. The quantization, or the existence of only certain allowed energies, arises when the particle is not free but is trapped by a potential, such as an electron trapped in an atom. Since trapped particles are very important to scientists, the entire subject is called "quantum" science.
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 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.