Vectors and operators in a Hilbert space are independent quantities and there is no obvious link between them. The mathematical description of the states and observables of a quantum system given by Postulates 25.1(PS) and 26.1(OV) in terms of vectors and operators does not tell us how a given state would determine the value of an observable. Physically the relationship between states and observables is about how the measurable values of an observable is related to a state. This relationship should lead to the quantum properties QMP5.3(1) to QMP5.3(3). First we must know how a state can give rise to a probability distribution of measurable values of an observable. This would require a prescription to relate a unit vector to the spectrum of selfadjoint operators. The discussion in Chapter 22 tells us how to obtain such a prescription. Because of the intricacy of the relationship both physically and mathematically we shall set out the relationship for discrete and continuous observables separately.
Vectors and operators in a Hilbert space are independent quantities and there is no obvious link between them. The mathematical description of the states and observables of a quantum system given by Postulates 25.1(PS) and 26.1(OV) in terms of vectors and operators does not tell us how a given state would determine the value of an observable. Physically the relationship between states and observables is about how the measurable values of an observable is related to a state. This relationship should lead to the quantum properties QMP5.3(1) to QMP5.3(3). First we must know how a state can give rise to a probability distribution of measurable values of an observable. This would require a prescription to relate a unit vector to the spectrum of selfadjoint operators. The discussion in Chapter 22 tells us how to obtain such a prescription. Because of the intricacy of the relationship both physically and mathematically we shall set out the relationship for discrete and continuous observables separately.
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.
The Hilbert-Space Description By the static description of a physical system we mean the rules that assign specified mathematical objects to the states and to the physical quantities of the system, and the prescriptions for calculating the probability distribution of the possible values of every physical quantity when the state of the system is given. In the usual Hilbert-space formulation of quantum mechanics, to each physical system is attached a separable Hilbert space (generally infinite-dimensional) over the complex field. To every physical quantity is associated a linear, self-adjoint, not necessarily bounded operator on. If one deals with a strictly quantum system, then the converse is also generally assumed: every self-adjoint operator on represents some physical quan- tity. The restriction “strictly quantum” is necessary: if the system retains some nonquantum (i.e. classical) feature, so that it requires the algorithm of so-called “superselection rules” (see Chapter 5), then there are self-adjoint operators on that do not represent physical quantities. It should also be stressed that even in the strictly quantum case, most of the self-adjoint operators actually do not represent “interesting” physical quantities: only a few of them represent physical quantities that are useful and meaningful for the description of the physical system (e.g., energy, momentum, position, angular momentum). Therefore, by asserting that every self-adjoint operator on corresponds to some physical quantity we mean there is no a priori impossibility of devising such a correspondence, but we do not claim that this correspondence is present in the real work of experimental physics.
Abstract This chapter develops the fundamental notions and concepts of the probabilistic and statistical interpretation of quantum mechanics. It begins with a brief review of the mathematical structure given by a Hilbert space of state vectors and an algebra of observables represented by selfadjoint operators. Special emphasis is laid on the connection between the spectral representation of a selfadjoint operator and a corresponding random variable that describes the possible measurement outcomes. The chapter further introduces the concepts of a composite quantum system, of the reduced density matrix, and of entangled quantum states. A section on quantum entropies is included, as well as the mathematical formulation of the generalized theory of quantum measurements in terms of completely positive quantum operations and effects.
The state of a quantum mechanical system is represented by a nonzero vector in a separable Hilbert space. The word ‘represented’ means that knowledge of the state vector gives complete information on the properties of the system. In quantum mechanics, the state vectors are denoted using Dirac's notation. Several quantum systems possess a property called spin. Spin is a purely quantum mechanical feature which has no direct classical counterpart. The set of density matrices is a convex subset of the space of Hermitian operators whose ‘extreme’ points represent pure states. In quantum mechanics, observed quantities are called observables. A continuous spectrum is such that it contains no isolated points. The chapter discusses the case of measurement of observables with discrete spectrum.
Many selfadjoint operators important to physical applications, e.g., number operators and some first order and second order differential operators, possess a discrete spectrum. For a selfadjoint operator with a discrete spectrum the spectral function reduces to a sum in the form. An arbitrary symmetric operator does not possess a spectral function and the spectral theorem does not apply to such an operator. Hence the relation between selfadjoint operators and probability distributions presented above does not apply to symmetric operators. This is the reason that the probabilistic nature of quantum mechanics is formulated in terms of selfadjoint operators, not symmetric operators. In the momentum representation the momentum appears as a multiplication operator in the same way that the position acts as a multiplication operator in the coordinate representation. It follows that we can treat the momentum in the momentum representation in the same way as position in the coordinate representation.
We review the postulates of quantum mechanics with respect to the representation of physical states and measurable quantities, their time evolution, and the interpretation of measurements. We first formulate the postulates in terms of wave functions and differential operators, and then reformulate them in the abstract Hilbert space of state vectors, using Dirac’s notations. Improper states subject to Dirac’s delta normalization are introduced, and the space of physical states is extended to include them. The postulates are rationalized by associating each Hermitian linear operator with a complete orthonormal system of its eigenvectors, where measurement probabilities depend on the projections of these eigenvectors on the system’s state vector. Particularly, wave functions are identified as projections of state vectors on the position operator eigenstates. State vectors representing multidimensional systems are formulated as tensor products of vectors in their subspaces. Finally, we address the general uncertainty relations in simultaneous measurements of different observables.