The form of Schrödinger's equation in a continuous representation is indicated for general systems and analyzed in detail for elementary Bose and Fermi systems for which illustrative solutions are given. For any system, a natural continuous representation exists in which state vectors are expressed as continuous, bounded functions of the corresponding classical variables. The natural continuous representation is generated by a suitable set 𝔖 of unit vectors labeled by classical variables for which, for the system in question, the quantum action functional restricted to the domain 𝔖 is equivalent to the classical action. When a classical action is viewed in this manner it contains considerable information about the quantum system. Augmenting the classical action with some physical significance of its variables, we prove that the classical theory virtually determines the quantum theory for the Bose system, while it uniquely determines the quantum theory for the Fermi system.
The form of Schrödinger's equation in a continuous representation is indicated for general systems and analyzed in detail for elementary Bose and Fermi systems for which illustrative solutions are given. For any system, a natural continuous representation exists in which state vectors are expressed as continuous, bounded functions of the corresponding classical variables. The natural continuous representation is generated by a suitable set 𝔖 of unit vectors labeled by classical variables for which, for the system in question, the quantum action functional restricted to the domain 𝔖 is equivalent to the classical action. When a classical action is viewed in this manner it contains considerable information about the quantum system. Augmenting the classical action with some physical significance of its variables, we prove that the classical theory virtually determines the quantum theory for the Bose system, while it uniquely determines the quantum theory for the Fermi system.
This paper discusses an application to the study of dynamics of the typical overcomplete, non-independent sets of unit vectors that characterize continuous-representation theory. It is shown in particular that the conventional, classical Hamiltonian dynamical formalism arises from an analysis of quantum dynamics restricted to an overcomplete, nonindependent set of vectors which lie in one-to-one correspondence with, and are labeled by, points in phase space. A generalized ``classical'' mechanics is then defined by the extremal of the quantum-mechanical action functional with respect to a restricted set of unit vectors whose c-number labels become the dynamical variables. This kind of ``classical'' formalism is discussed in some generality, and is applied not only to simple single-particle problems, but also to finite-spin degrees of freedom and to fermion field oscillators. These latter cases are examples of an important class of problems called exact, for which a study of the classical dynamics alone is sufficient to infer the correct quantum dynamics.
In a continuous representation of Hilbert space, each vector ψ is represented by a complex, continuous, bounded function ψ(φ) ≡ (φ, ψ) defined on a set S of continuously many, nonindependent unit vectors φ having rather special properties: Each vector in S possesses an arbitrarily close neighboring vector, and the identity operator is expressable as an integral over projections onto individual vectors in S. In particular cases it is convenient to introduce labels for the vectors in S whereupon each ψ is represented by a complex, continuous, bounded, label-space function. Basic properties common to all continuous representations are presented, and some applications of the general formalism are indicated.
This chapter discusses in some detail the connection between the classical and quantum mechanical equations of motion, largely using examples that are solvable. It recounts how the old quantum theory was formulated by quantizing the action. Choosing irreducible cycles around the tori leads naturally to defining the action and angle variables of motion, and the quantization of action around each cycle. The chapter shows that families of these periodic orbits contribute to the oscillating part of the density of states of an integrable system in the form of a trace formula. It discusses the transition from integrability to chaos in such systems when perturbations are added that couple resonantly to these natural frequencies. The chapter elaborates with a view to develop the semiclassical approximation, and the associated quantization condition. It brings out the similarity between the Schrodinger equation of wave mechanics, and the HJ equation.
The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.
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.
This paper is intended to show the Schrodinger equation, within its structure, allows the manifestation of the wave function collapse within a very natural way of reasoning. In fact, as we will see, nothing new must be inserted to the classical quantum mechanics, viz., only the dialectics of the physical world must be interpreted under a correct manner. We know the nature of a physical system turns out to be quantical or classical, and, once under the validity of the Schrodinger equation to provide the evolution of this physical system, the dialectics, quantum or classical, mutually exclusive, must also be under context through the Schrodinger equation, issues within the main scope of this paper. We will show a classical measure, the obtention of a classical result, emerges from the structure of the Schrodinger equation, once one demands the possibility that, over a chronological domain, the system begins to provide a classical dialectic, showing the collapse may be understood from both: the structure of the Schrodinger equation as well as from the general solution to this equation. The general solution, even with a dialectical change of description, leads to the conservation of probability, obeying the Schrodinger equation. These issues will turn out to be a consequence of a general potential energy operator, obtained in this paper, including the possibility of the classical description of the physical system, including the possibility of interpretation of the collapse of the quantum mechanical state vector within the Schrodinger equation scope.