Based on classical mechanical picture of physical objects, quantum-mechanical measurements are formulated and analyzed in an abstract way. We interpret a dynamical flow generated by a Hamiltonian vector field as a description of the effect on the objects caused by a device or an environment. By expressing contexts of measurement explicitly, a matrix representation of observables on a finite dimensional Hilbert space is constructed. A quantum-mechanical state vector is introduced as something on which a dynamical flow generated by a Hamiltonian vector field acts. We find conditions that enable us to obtain the Dirac's quantization rule for the mean values. At the same time, several characteristics (i.e., rectifiability, a generator of symmetry, an invariant mean) of quantum-mechanical measurements are formulated in our formalism.
Based on classical mechanical picture of physical objects, quantum-mechanical measurements are formulated and analyzed in an abstract way. We interpret a dynamical flow generated by a Hamiltonian vector field as a description of the effect on the objects caused by a device or an environment. By expressing contexts of measurement explicitly, a matrix representation of observables on a finite dimensional Hilbert space is constructed. A quantum-mechanical state vector is introduced as something on which a dynamical flow generated by a Hamiltonian vector field acts. We find conditions that enable us to obtain the Dirac's quantization rule for the mean values. At the same time, several characteristics (i.e., rectifiability, a generator of symmetry, an invariant mean) of quantum-mechanical measurements are formulated in our formalism.
Quantum mechanics is based on Schrodinger’s wave function with linear superposition of vectors in a Hilbert space. Due to superposition, multiple physical states are considered simultaneously for the same time point, but when the corresponding quantum mechanical experiments are realized, there is only one outcome and not a superposition of pointer positions. This observation characterizes the quantum mechanical measurement problem, which for about a century found multiple interpretations, which can be classified as potentiality, as proposed by Heisenberg, or as real physical entities, for instance fields. Potentialities are yet inexistent in reality and unobservable in the present, since they represent a possible future. Thus, they can only be an intra-mental representation of a future process and not a physical extra-mental process ongoing in the present. There is coexistence of a physical object observable in reality and a mental prediction by an observer concerning the future behavior of this object. The characterization of quantum mechanical formalism by some physicists corresponds to a prediction and therefore to a mental representation of the future behavior of elementary particles and not to a description of an ongoing extra-mental process. Prediction in life and science has some characteristics, which resemble the measurement problem, since it undergoes a “multiple to one” reduction of mental potentialities to only one observable real outcome. Besides quantum mechanics, such reductions can also be found with many kinds of predictions, in classical physics for regular and irregular dynamical processes and in humans for programming future actions. Superposition in classical physics only concerns variable effects, but in quantum mechanics includes variable effects and non-localizable causes. Quantum mechanics, interpreted as prediction of future outcomes with multiple potentialities but only one realization, would resemble the general prediction problem in life and science and thereby loos its weird aspects.
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.
Classical mechanics is usually valid at the macro scale. The changes in the position and the momentum of the object over time are described by the Hamiltonian equation of motion. The wave function in quantum mechanics, if unobserved, evolves in a smooth and continuous way according to the Schrodinger equation, which is related to the Hamiltonian equation of motion. This equation describes a linear superposition of different states at time. A quantum mechanical description of a physical system is related to a probabilistic representation; it is described by a vector in Hilbert space. This description extends the two- or three-dimensional Euclidean space into spaces that have any finite or infinite number of dimensions. The state of a compound system is projected to the subspace that corresponds to the observed state and the vector representing the state is renormalized to the unit length. An observable describes a subspace for some dimensions with a special case of one dimension.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
In the absence of a satisfactory interpretation of quantum theory, physical law lacks physical basis. This paper reviews the orthodox, or Dirac–von Neumann interpretation, and makes explicit that Hilbert space describes propositions about measurement results. Kets are defined as conditional clauses referring to measurements in a formal language. It is seen that these clauses are elements of a Hilbert space, such that addition is logical disjunction, the dual space consists of consequent clauses, and the inner product is a set of statements in the subjunctive mood. The probability interpretation gives truth values for corresponding future tense statements when the initial state is actually prepared and the final state is to be measured. The mathematical structure of quantum mechanics is formulated in terms of discrete measurement results at finite level of accuracy and does not depend on an assumption of a substantive, or background, space–time continuum. A continuum of kets, ∣x〉 for x ∈ ℝ3, is constructed from linear combinations of kets in a finite basis. The inner product can be expressed either as a finite sum or as an integral. Discrete position functions are uniquely embedded into smooth wave functions in such a way that differential operators are defined. It is shown that the choice of basis has no effect on underlying physics (quantum covariance). The Dirac delta has a representation as a smooth function. Operators do not in general have an integral form. The Schrödinger equation is shown from the requirements of the probability interpretation. It is remarked that a formal construction of quantum electrodynamics (qed) avoiding divergence problems has been completed using finite dimensional Hilbert space. I conclude that quantum mechanics makes statements about the world with clear physical meaning, such that space is emergent from particle interactions and has no fundamental role.
This paper is concerned with the description of the process of measurement within the context of a quantum theory of the physical world. It is noted that quantum mechanics permits a quasi-classical description (classical in the limited sense implied by the correspondence principle of Bohr) of those macroscopic phenomena in terms of which the observer forms his perceptions. Thus, the process of measurement in quantum mechanics can be understood on the quasi-classical level by transcribing from the strictly classical observables of Newtonian physics to their quasi-classical counterparts the known rules for the measurement of the former. The remaining physical problem is the delineation of the circumstances in which the correlation of a peculiarly quantum mechanical observable A with a classically measurable observable B can result in a significant measurement of A. This is undertaken within the context of quantum theory. The resulting clarification of the process of measurement has important implications relative to the philosophic interpretation of quantum mechanics.