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.
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.
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.
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.
Although• quantum mechanics has achieved great success in the explanation of physical phenomena, the mathematics used in quantum mechanics is not rigorous.Unrigorous treatments occur in the case of continuous spectrum.In a Hilbert • space, let A be a self-adjoint operator.When we define the eigenvalue and the eigenelement by the ' ' relation Af=J.f, (1) ' ' h d -. -f(q)=).f(q)2m dq has solutions 2,ri f(q)=ce"Aq for any ;.., But these are not quadratically integrable.E. Hellinger and H. Weyi< 0 have already investigated the problem of continuous spectrum and introduced the " eigendifferential."But they are concerned with a special problem, i. e. bounded quadratic forms and solutions of differential equation.In respect of generality, they do not satisfy.< 2 > E. Fues< 3 > and some other writers<" have used this conception " eigendifferential " in quantum mechanics.But the treatments are either complicated or not rigorous.J. v. Neumann< 5 > has proved that in Hilbert space, for any selfadjoint operator A, there exists a resolution of identity E(;..) such that Af= J:and he has applied this theory to quantum mechanics.Although his theory gave a rigorous mathematical foundation to quantum mechanics, it did not give any interpretation to the representation theory of Dirac's form.Recently I investigated the "theory of vector-valued set functions."This theory removes the difficulties in the treatement of continuous spectrum, and gives a rigorous interpretation to the representation theory.< 5 > The vector-valued set function q(U) is a function of set U, and its functional value is an element (vector) in the Hilbert ,space, (1) E. Hellinger [l], 231-271.H. Weyl [1], 238-251.(2) T. Carlemann used this conception "eigendifferential" in the treatise of singular integral equations.Cf.T. Carlemann [1].(3) E. Fues [1], 295-303.(4) For example, L. de Broglie [2], 140-159.(5) J. v. Neumann [1], 53-121.(6) F. Maeda [2], [5], [6].A summary of the theory of vector-valued set functions is given in the first part Qf the present paper.(1) G. Temple has formulated a principle of quantum theory in which projective operators are given priority over self-adjoint operators.(C(.G. Temple [1)).But. the .whole discussion differs from that of my present paper.
There are of course many different ways of generating probabilities. In many classical statistical experiments we can assume all outcomes are equally likely to occur. This chapter introduces different ways to produce the required probability distribution. Generally the probability of each of measured outcome would depend on the state of the system as well as that particular observable in question. A simple counting of number of outcomes without reference to the state cannot possibly lead to the correct probability distribution. The theory presented can be extended to lower as well as higher dimensional real vector spaces. For a model theory to have any chance of success it is necessary for the model to satisfy the following additional requirements: There must be enough unit vectors and selfadjoint operators to correspond to all the states and all the observables of the system; and the probability distribution for every observable in every state must agree with experimental results.
A simple operation is to lengthen or to shorten a vector by a factor a without changing its orientation. The simplest operator of this kind is the identity operatorII^. The operators introduced so far have a clear geometric meaning. One should use these operators as building blocks to establish other operators which may not have distinctive geometric properties. The building blocks we have in mind are projectors. Selfadjoint matrices and selfadjoint operators have many distinctive properties which are crucial to the probabilistic formulation of quantum mechanics. Since selfadjoint operators are represented by selfadjoint matrices one can solve their eigenvalue problem in terms of their representative selfadjoint matrices to obtain similar results, e.g., the contents of Theorems for selfadjoint matrices also carry over to selfadjoint operators. Selfadjoint operators with non-negative eigenvalues can be singled out to form a distinctive group with many useful properties.
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.