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Classical Limit on Quantum Mechanics for Unbounded Observables

Pun Wai Tong

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

This dissertation is divided into two parts. In Part I of this dissertation--- On the Classical Limit of Quantum Mechanics, we extend a method introduced by Hepp in1974 for studying the asymptotic behavior of quantum expectations in the limitas Plank's constant ($\\hbar$) tends to zero. The goal is to allow forunbounded observables which are (non-commutative) polynomial functions of theposition and momentum operators. [This is in contrast to Hepp's original paperwhere the "observables" were, roughly speaking, required to be bounded functionsof the position and momentum operators.] As expected the leading ordercontributions of the quantum expectations come from evaluating the "symbols" of the observables along the classical trajectories while the next order contributions (quantumcorrections) are computed by evolving the $\\hbar=1$ observables by a linear canonicaltransformations which is determined by the second order pieces of the quantum mechanical Hamiltonian. Part II of the dissertation --- Powers of Symmetric Differential Operators is devoted to operator theoretic properties of a class of linear symmetric differential operators on the real line. In more detail, let $L$ and $\\tilde{L}$ be a linear symmetric differential operator with polynomial coefficients on $L^{2}\\left(m \\right) $ whose domain is the Schwartz test function space, $\\mathcal{S}.$ We study conditions on the polynomial coefficients of $L$ and $\\tilde{L}$ which implies operator comparison inequalities of the form $\\left( \\overline{\\tilde{L}}+\\tilde{C}\\right) ^{r}\\leq C_{r}\\left( \\bar{L}+C\\right) ^{r}$for all $0\\leq r<\\infty.$ These comparison inequalities (along with their generalizationsallowing for the parameter $\\hbar>0$ in the coefficients) are used to supply alarge class of Hamiltonian operators which verify the assumptions needed forthe results in Part I of this dissertation.

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This dissertation is divided into two parts. In Part I of this dissertation--- On the Classical Limit of Quantum Mechanics, we extend a method introduced by Hepp in1974 for studying the asymptotic behavior of quantum expectations in the limitas Plank's constant (hbar\\hbarhbar) tends to zero. The goal is to allow forunbounded observables which are (non-commutative) polynomial functions of theposition and momentum operators. [This is in contrast to Hepp's original paperwhere the "observables" were, roughly speaking, required to be bounded functionsof the position and momentum operators.] As expected the leading ordercontributions of the quantum expectations come from evaluating the "symbols" of the observables along the classical trajectories while the next order contributions (quantumcorrections) are computed by evolving the hbar=1\\hbar=1hbar=1 observables by a linear canonicaltransformations which is determined by the second order pieces of the quantum mechanical Hamiltonian. Part II of the dissertation --- Powers of Symmetric Differential Operators is devoted to operator theoretic properties of a class of linear symmetric differential operators on the real line. In more detail, let LLL and tildeL\\tilde{L}tildeL be a linear symmetric differential operator with polynomial coefficients on L2left(mright)L^{2}\\left(m \\right) L2left(mright) whose domain is the Schwartz test function space, mathcalS.\\mathcal{S}.mathcalS. We study conditions on the polynomial coefficients of LLL and tildeL\\tilde{L}tildeL which implies operator comparison inequalities of the form left(overlinetildeL+tildeCright)rleqCrleft(barL+Cright)r\\left( \\overline{\\tilde{L}}+\\tilde{C}\\right) ^{r}\\leq C_{r}\\left( \\bar{L}+C\\right) ^{r}left(overlinetildeL+tildeCright)rleqCr​left(barL+Cright)rfor all 0leqr<infty.0\\leq r<\\infty.0leqr<infty. These comparison inequalities (along with their generalizationsallowing for the parameter hbar>0\\hbar>0hbar>0 in the coefficients) are used to supply alarge class of Hamiltonian operators which verify the assumptions needed forthe results in Part I of this dissertation.
This paper is devoted to the study of the classical limit of quantum mechanics. In more detail we will elaborate on a method introduced by Hepp in 1974 for studying the asymptotic behavior of quantum expectations in the limit as Plank's constant (ℏ)\hbar)ℏ) tends to zero. Our goal is to allow for unbounded observables which are (non-commutative) polynomial functions of the position and momentum operators. This is in contrast to Hepp's original paper where the observables were, roughly speaking, required to be bounded functions of the position and momentum operators. As expected the leading order contributions of the quantum expectations come from evaluating the observables along the classical trajectories while the next order contributions are computed by evolving the ℏ=1\hbar=1ℏ=1 observables by a linear canonical transformations which is determined by the second order pieces of the quantum mechanical Hamiltonian.
The thesis divides naturally into two parts. Part I raises, and in some cases answers, questions concerning symmetry in classical mechanics. The main result (Theorem 6.4) shows that the assumption of the existence of a realization puts an upper limit on the rank of the algebra. \n \n The heart of the thesis (covering three-quarters of the volume) is section II on the quantization of classical systems. §1 lists axioms desirable in any quantization rule for the 'functions of the q's'. The momentum observables are introduced in §2 prior to their quantization in §4. §5 essentially shows how conventional quantum mechanics fits into this scheme of things. By progressive specialization from a general manifold to a vector space, from a general quantization scheme to one which is linear on the linear momentum functions, and finally to an entirely well-behaved (admissible) quantization rule, into which conventional quantum mechanics fits nicely, we obtain in §7-§9 results which become progressively more and more powerful. The final theorem (Theorem 9.2) is perhaps the most significant of all. This result states that there exists a class of functions, which contains all functions of the q's and functions of the p's and all momentum observables and which is closed with respect to any linear canonical transformation L; a rule A assigning a unique self-adjoint operator to each such function f; a unitary operator WL corresponding to L and an equation \n \n𝛬(𝑓 ∘ 𝐿) = 𝑊[sub]𝐿⁻ 𝛬 𝑓 𝑊[sub]𝐿
The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing h0h_0h0​. For the second of these two extreme values, introduced operatorial algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operatorial framework.
Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.
This dissertation investigates statistical constraints on measurement outcomes in quantum mechanics, addressing two fundamental features that distinguish quantum theory from classical physics: non-reality and uncertainty.We examine these features through two perspectives: observable-dependent statistical constraints on quantum correlations and estimationbased measurement uncertainty in infinite-dimensional quantum systems.In the first part, we study realizability problems in both local realism and quantum theory, focusing on observable-dependent constraints.While previous studies have mainly characterized quantum correlations independently of observables, we derive a necessary and sufficient condition for quantum realizability of correlations in the Clauser-Horne-Shimony-Holt (CHSH) scenario when observables are fixed in advance.Our main result provides an explicit inequality that imposes stronger constraints than previously known observable-independent bounds.We demonstrate that the maximum CHSH inequality violation depends on observable choice and apply our results to Bell's original scenario, revealing that whether Bell's original inequality is violated depends sensitively on observable selection.These findings demonstrate that non-reality does not manifest equally in all measurement configurations.In the second part, we extend the estimation-based formulation of measurement uncertainty by Watanabe, Sagawa, and Ueda to infinite-dimensional quantum systems.The original formulation provided clear physical interpretation but excluded fundamental observables such as position and momentum.Using pseudo-inverse forms of positive-semidefinite sesquilinear forms, we develop classical and quantum estimation theory for statistical models whose parameter space is the set of full-rank quantum states on infinite-dimensional Hilbert spaces.We derive classical and quantum Cramér-Rao inequalities and define estimationbased measurement errors for both bounded and unbounded operators.For each error type, we establish error-error uncertainty relation inequalities stronger than the original Watanabe-Sagawa-Ueda bound due to an improved derivation utilizing the full Cauchy-Schwarz inequality.Our work provides a universal definition of measurement errors with a clear physical interpretation while maintaining mathematical rigor.Our results contribute to quantum foundations by deepening the understanding of how measurement outcome constraints arise from quantum structure and observable properties.The methodological tools introduced-including characteristic functions, quasi-joint probability distributions, symmetry-based reduction techniques, and pseudo-inverse forms-may prove useful for future investigations in quantum foundations and quantum information theory.
For a quantum observable AℏA_\hbarAℏ​ depending on a parameter ℏ\hbarℏ we define the notion ``AℏA_\hbarAℏ​ converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that Aℏ→0A_\hbar\to0Aℏ​→0 is equivalent with ∥Aℏ∥→0\Vert A_\hbar\Vert\to0∥Aℏ​∥→0. The ℏ\hbarℏ-wise product of convergent observables converges to the product of the limiting phase space functions. ℏ−1\hbar^{-1}ℏ−1 times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the ℏ\hbarℏ-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.