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Reviewing Observables in Classical and Quantum Mechanics

Roshan Raj

Published Oct 10, 2023• 0 citations
Publisher

Abstract

It presents the evolution of the quantum observables from their classical counterpart and distinctly analyses the meaning of observable in context to both the classical and the quantum realms, with an emphasis on W. Heisenberg's realization of observable and unobservable [1925]. It is stated that quantum observables are, in the first place, indeed distinctive to classical observables since each appears unobservable to the other; however, to associate physical essence as possessed by the latter, the expectation value of the former is determined. Further, the possibility of mapping quantum information to the classical ones through the action of the operator may be realized after re-interpreting analogies between the quantum commutator and Poisson brackets.

DOI
10.31219/osf.io/p2ufx

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It presents the evolution of the quantum observables from their classical counterpart and distinctly analyses the meaning of observable in context to both the classical and the quantum realms, with an emphasis on W. Heisenberg's realization of observable and unobservable [1925]. It is stated that quantum observables are, in the first place, indeed distinctive to classical observables since each appears unobservable to the other; however, to associate physical essence as possessed by the latter, the expectation value of the former is determined. Further, the possibility of mapping quantum information to the classical ones through the action of the operator may be realized after re-interpreting analogies between the quantum commutator and Poisson brackets.
This chapter explores the Schrödinger and Heisenberg pictures in quantum mechanics, emphasizing their mathematical equivalence and physical consistency. The two formulations are shown to be connected by a unitary transformation. The Heisenberg equation of motion is derived, and the time evolution of operators—particularly those not explicitly dependent on time—is expressed in terms of their initial values. It is demonstrated that scalar products, expectation values, matrix elements, and commutation relations remain invariant under the change of picture. Drawing a close analogy with classical mechanics, Dirac's insight is presented: the quantum mechanical counterpart of the classical Poisson bracket is the operator commutator divided by iℏ . The chapter concludes by deriving the Lorentz force law within the quantum framework using commutator algebra. Metadata/Keywords: Schrödinger picture, Heisenberg picture, unitary transformation, Heisenberg equation of motion, Poisson bracket, Lorentz force, time evolution, expectation values
In this work we discuss the notion of observable - both quantum and classical - from a new point of view. In classical mechanics, an observable is represented as a function (measurable, continuous or smooth), whereas in (von Neumann's approach to) quantum physics, an observable is represented as a bonded selfadjoint operator on Hilbert space. We will show in the present part II and the forthcoming part III of this work that there is a common structure behind these two different concepts. If R\mathcal{R}R is a von Neumann algebra, a selfadjoint element A∈RA \in \mathcal{R}A∈R induces a continuous function fA:Q(P(R))→Rf_{A} : \mathcal{Q}(\mathcal{P(R)}) \to \mathbb{R}fA​:Q(P(R))→R defined on the \emph{Stone spectrum} Q(P(R))\mathcal{Q}(\mathcal{P(R)})Q(P(R)) (\cite{deg3}) of the lattice P(R)\mathcal{P(R)}P(R) of projections in R\mathcal{R}R. fAf_{A}fA​ is called the observable function corresponding to AAA. The aim of this part is to study observable functions and its various characterizations.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
We analyze the difference between classical mechanics and quantum mechanics. We come to the conclusion that this difference can be found in the nature of the observables that are considered for the physical system under consideration. Classical mechanics can only describe a certain kind of what we called ‘‘classical observable.’’ Quantum mechanics can only describe another kind of observable; it cannot describe, however, classical observables. To perform this analysis, we use a theory where every kind of observable can be treated and which is in a natural way a generalization of both classical and quantum mechanics. If in a study of a physical system in this theory we restrict ourselves to the classical observables, we rediscover classical mechanics as a kind of first study of the physical system, where all the nonclassical properties are hidden. If we find that this first study is too rough we can also study the nonclassical part of the physical system by a theory which is eventually quantum mechanics.
The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical observables are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest’s theorem is shown to be Lie–Poisson for a semidirect-product Lie group, named the Ehrenfest group . The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie–Poisson structure associated with another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models that have previously appeared in the chemical physics literature.