According to the postulates of quantum mechanics, the state of a system is associated with a wave function that contains any measurable information on the system at any time. In this chapter we become familiar with wave functions and how they represent the position of particles within the system. Within the realm of quantum mechanics, the position of particles is not deterministic. It is defined by a probability distribution. The wave function is a position-dependent complex-valued amplitude, whose absolute value squared is identified with the probability density for locating the particle in the position space. This identification of the wave function with a probability amplitude imposes some limitation. Particularly, for a closed system in which the particles are bound, the wave function must be proper (square integrable) and normalizable. These properties are discussed and demonstrated for different coordinate systems.
Ever since the beginning of quantum mechanics, the measurement problem has been a subject of sometimes discontinued but nevertheless recurrent concern. As students, most physicists have raised questions about it, but the spectacular and renewed successes of quantum theory were a strong incentive for them to work out applications without further worrying about foundations. In each generation, however, some dissatisfaction reemerges, which leads to slow progress, and we may expect that the measurement problem will some day be cleared up rather than forgotten. The situation recalls statistical mechanics, the foundations of which still give rise to scientific discussions, in spite of more than one century’s achievements. Actually, the paradoxes of irreversibility and of Maxwell’s demon share some common features with quantum measurements, to wit, they involve probabilities in a crucial fashion and their explanation requires elucidating the role of the observer.
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function
ψ. Given a Schrodinger wave function
ψ, we use its value
ψ0 at some fixed time (say,
t=0) to define the probability measure
∣ψ0∣2dx on the system configuration space
M (
=Rn). On the resulting probability space
M0, we introduce a stochastic process
ξ(t) corresponding to the Heisenberg position operator
XH(t) such that, in the Heisenberg state
∣ψh⟩ corresponding to
ψ0, the expectation value of
XH(t) equals that of
ξ(t) in
M0. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process
ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the
ψ0-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.
Based on the Chetaev theorem on stable dynamical trajectories in the presence of dissipative forces, we obtain the generalized condition for stability of relativistic classical Hamiltonian systems (with an invariant evolution parameter) in the form of the Stückelberg equation. As is known, this equation is the basis of a competing paradigm known as parametrized relativistic quantum mechanics (pRQM). It is shown that the energy of dissipative forces, which generate the Chetaev generalized condition of stability, coincides exactly with Bohmian relativistic quantum potential. Within the framework of Bohmian RQM supplemented by the generalized Chetaev theorem and on the basis of the principle of least action for dissipative forces, we show that the squared amplitude of a wave function in the Stückelberg equation is equivalent to the probability density function for the number of particle trajectories, relative to which the velocity and the position of the particle are not hidden parameters. The conditions for reasonableness of trajectory interpretation of pRQM are discussed. Based on analysis of a general formalism for vacuum-flavor mixing of neutrino within the context of the standard and pRQM models we show that the corresponding expressions for the probability of transition from one neutrino flavour to another differ appreciably, but they are experimentally testable: the estimations of absolute value for neutrino mass based on modern experimental data for solar and atmospheric neutrinos show that the pRQM results have a preference. It is noted that the selection criterion of mass solutions relies on proximity between the average size of condensed neutrino clouds, which is described by the Muraki formula (29th ICRC, 2005) and depends on the neutrino mass, and the average size of typical observed void structure (dark matter + hydrogen gas), which plays the role of characteristic dimension of large-scale structure of the Universe.