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InTech eBooks

Correspondence, Time, Energy, Uncertainty, Tunnelling, and Collapse of Probability Densities

Gabino Torres-Vega

Published Feb 24, 2012• 4 citations
Publisher

Abstract

Classical and Quantum Mechanics make use of different objects and they also were written on different background spaces.As a consequence, they have followed different paths regarding the mathematical objects that they use.Classical Mechanics makes use of points and functions on a real cotangent space T * Q of a mechanical system, and makes use of differential geometry as the basic language with which the theory is developed.This is due to the existence of trajectories of single points.On the other side, Quantum Mechanics makes use of state vectors in a complex Hilbert space, with operators, commutators, and eigenvectors, and makes use of some postulates that look weird from a classical point of view.This is a point of view that was induced by the lack of trajectories and by the use of probabilistic interpretations of state vectors.There have been efforts to define similar classical and quantum functions that can be compared with each other.Quantum densities were written as functions on phase-space by means of integral transformations.Two of these transformations are the Wigner (

DOI
10.5772/32996

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Classical and Quantum Mechanics make use of different objects and they also were written on different background spaces.As a consequence, they have followed different paths regarding the mathematical objects that they use.Classical Mechanics makes use of points and functions on a real cotangent space T * Q of a mechanical system, and makes use of differential geometry as the basic language with which the theory is developed.This is due to the existence of trajectories of single points.On the other side, Quantum Mechanics makes use of state vectors in a complex Hilbert space, with operators, commutators, and eigenvectors, and makes use of some postulates that look weird from a classical point of view.This is a point of view that was induced by the lack of trajectories and by the use of probabilistic interpretations of state vectors.There have been efforts to define similar classical and quantum functions that can be compared with each other.Quantum densities were written as functions on phase-space by means of integral transformations.Two of these transformations are the Wigner (
In the standard formulation of quantum mechanics, observables are represented by operators in a Hilbert space and states by vectors in it. Quantum field theory too is formulated in terms of field operators. States in field theory include the vacuum state, one particle states, two particle states and so on. One does not measure operators or state vectors, however; measurements involve eigenvalues of operators or transition probabilities between states. It turns out that one can formulate quantum mechanics and field theory without using operators. Indeed, the probability amplitude for a transition from a position q1 at time t1 to a position q2 at time t2 can be expressed as a path integral ∫ dt L(q(t),q˙(t)), (5.1) where q(t) runs over all possible functions satisfying the conditions q(t1) = q1, q(t2) = q2 (5.2) and the integral over q(t) is an integral over all such paths of the exponential evaluated for each path. The measure of integration is not well defined a priori, but it is possible to give prescriptions that make sense. In classical mechanics, positions at two different times determine a path from the second order differential equation of motion, which yields a definite solution if two conditions are imposed. In contrast, quantum mechanics does not allow us to talk about trajectories. The path integral formulation manifestly refers to the myriad paths that may be taken by a particle in going from one point to another. It goes so far as to assign a probability amplitude to each path, and it is related to the classical action corresponding to that path. However, it is complex and it also involves a superposition, characteristic of quantum mechanics.
In classical mechanics the state of a system is defined by the numerical values of a set of dynamical variables such as the position coordinates and the velocities. In quantum mechanics the state of a system at an instant is represented by a unit vector in a Hilbert space, where a set of axes are the eigenvectors of a complete set of observables. The description of quantum dynamics where the wave function is time-dependent and the operators describing the observables are time-independent is known as the Schrodinger picture. The use of total time derivative of operator that depends on the time evolution of the wave function as well as on any intrinsic time-dependence in the operators simplifies the development of quantum mechanics and allows the development to more closely follow the corresponding development of classical mechanics. The difference between the Schrodinger and Heisenberg pictures is best understood as the difference in the basic set of observables.
Basically, there are two approaches to the formulation of quantum mechanics, i.e., the operator and the Feynman path integral approach. The main object in quantum mechanics is referred to as the transition amplitude between those states, matrix elements of physical operators and, in particular, the S- matrix in the problem of quantum scattering. On the basis of quantum mechanics, the operator method is closed, well formulated mathematically, and so is an appropriate universal language describing quantum field theory, the quantum physics of condensed states and quantum relativistic statistical physics. The differential equation defining the path of the system in the configuration space is called the equation of motion of the system. The aim of classical as well as quantum mechanics is to predict the time evolution of the system, starting from knowledge of the state of the system at a given point in time.
This acceptance talk is a curious mixture of personal history and developing ideas in the context of the growing field of IR covering several decades. I want to concentrate on models and theories, interpreted loosely, and try and give an insight into where I have got to in my thinking, where the ideas came from, and where I believe we are going.In the last few years I have been working on the development of what might be coined as a design language for IR. It takes its inspiration from Quantum Mechanics, but by analogy only. The mathematical objects represent documents; these objects might be vectors (or density operators) in an n-dimensional vector space (usually a Hilbert space). A request for information, or a query, is taken as an observable and is represented as a linear operator on the space. Linear operators can be expressed as matrices. Such an operator, Hermitian, has a set of eigenvectors forming a basis for the space; which we interpret as a point of view or perspective from which to understand the space. Thus any document-vector can be located with respect to the basis, and we can calculate an inner product between such a vector and any basis vector, which may be interpreted as a probability of relevance. The probability of observing any given eigenvector is now given by the square of that inner product assuming all vectors are normalised. Hence we connect the probability of observation to the geometry of the space. Furthermore, the subspaces of the space make up a lattice structure which is equivalent to a logic. This makes up the entire mathematical structure, and the language for handling this structure is linear algebra: vectors, matrices, projections, inner-products, neatly captured by the Dirac notation used in quantum mechanics. Our probability is slightly different from classical probability, the same for logic; we end up with quantum logic and quantum probability.A commitment to this kind of mathematical structure, with which to model objects and processes in IR, depends on two critical assumptions.
In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function W(q,p)W(q,p)W(q,p), the marginals of which yield the correct quantum probabilities for qqq and ppp separately \cite{wigner}. Its usefulness stems from the fact that it provides a re-expression of quantum mechanics in terms of classical concepts so that quantum mechanical expectation values are now expressed as averages over phase-space distribution functions. In other words, statistical information is transferred from the density operator to a quasi-classical (distribution) function.