arXiv:1511.02447

On the classical limit of quantum mechanics I

Bruce K. Driver, Pun Wai Tong

Published Nov 8, 2015• 1 citation

Abstract

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 ($\hbar)$ 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 $\hbar=1$ observables by a linear canonical transformations which is determined by the second order pieces of the quantum mechanical Hamiltonian.

DOI: 10.48550/arxiv.1511.02447

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arXiv:1511.02447

On the classical limit of quantum mechanics I

Bruce K. Driver, Pun Wai Tong

Published Nov 8, 2015• 1 citation

arXiv HTML PDF

Abstract

\hbar)

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

\hbar=1

observables by a linear canonical transformations which is determined by the second order pieces of the quantum mechanical Hamiltonian.

DOI: 10.48550/arxiv.1511.02447

Related papers

Classical Limit on Quantum Mechanics for Unbounded Observables

Pun Wai Tong

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.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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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 $h_0$ . 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.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 $h_0$. 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.

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Modified Heisenberg Commutations Relations and Its Standard Hamiltonian Interpretation

Mauricio Contreras G., Roberto Ortiz, José Mauricio González

This paper analyzes the modified canonical Heisenberg commutation relations or GUP, from a standard Hamiltonian point of view. For a one-dimensional system, a such modified canonical Heisenberg commutation relation is defined by the commutator between a position x^ and a momentum operator p^ (called the deformed momentum), which becomes a function F of the same operators: x^,p^=F(x^,p^), that is, the Heisenberg algebra closes itself in general in a nonlinear way. The function F also depends on a parameter that controls the deformation of the Heisenberg algebra in such a way that for a null parameter value, one recovers the usual Heisenberg algebra x^,p^0=iℏI. Thus, it naturally raises the following questions: What does a relation of this type mean in Hamiltonian theory from a standard point of view? Is the deformed momentum the canonical variable conjugate to the position in such a relation? Moreover, what are the canonical variables in this model? The answer to these questions comes from the existence of two different phase spaces: The first one, called the non-deformed phase (which is obtained for control parameter value equal to zero), is defined by the Cartesian x^ coordinate and its non-deformed conjugate momentum p^0, which satisfies the standard quantum mechanical Heisenberg commutation relation. The second phase space, the deformed one, is given by the deformed momentum p^ and a new position coordinate y^, which is its canonical conjugate variable, so y^ and p^ also satisfy standard commutation relations. We construct a classical canonical transformation that maps the non-deformed phase space into the deformed one for a specific class of deformation functions F. Additionally, a quantum mechanical operator transformation is found between the two non-commutative phase spaces, which allows the Schrödinger equation to be written in both spaces. Thus, there are two equivalent quantum mechanical descriptions of the same physical process associated with a deformed commutation relation.This paper analyzes the modified canonical Heisenberg commutation relations or GUP, from a standard Hamiltonian point of view. For a one-dimensional system, a such modified canonical Heisenberg commutation relation is defined by the commutator between a position x^ and a momentum operator p^ (called the deformed momentum), which becomes a function F of the same operators: x^,p^=F(x^,p^), that is, the Heisenberg algebra closes itself in general in a nonlinear way. The function F also depends on a parameter that controls the deformation of the Heisenberg algebra in such a way that for a null parameter value, one recovers the usual Heisenberg algebra x^,p^0=iℏI. Thus, it naturally raises the following questions: What does a relation of this type mean in Hamiltonian theory from a standard point of view? Is the deformed momentum the canonical variable conjugate to the position in such a relation? Moreover, what are the canonical variables in this model? The answer to these questions comes from the existence of two different phase spaces: The first one, called the non-deformed phase (which is obtained for control parameter value equal to zero), is defined by the Cartesian x^ coordinate and its non-deformed conjugate momentum p^0, which satisfies the standard quantum mechanical Heisenberg commutation relation. The second phase space, the deformed one, is given by the deformed momentum p^ and a new position coordinate y^, which is its canonical conjugate variable, so y^ and p^ also satisfy standard commutation relations. We construct a classical canonical transformation that maps the non-deformed phase space into the deformed one for a specific class of deformation functions F. Additionally, a quantum mechanical operator transformation is found between the two non-commutative phase spaces, which allows the Schrödinger equation to be written in both spaces. Thus, there are two equivalent quantum mechanical descriptions of the same physical process associated with a deformed commutation relation.

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Mark L. Rosenquist, James W. Evans

The classical limit of quantum mechanics may be obtained, in a much simpler fashion than usual, by applying the de Broglie relation to Fermat’s principle.

American Journal of PhysicsOct 1, 19886 citations

A Formulation of Quantum Particle Swarm Optimization

Ichio Kikuchi, Akihito Kikuchi

In this article, we present our version of quantum particle swarm optimization. The objective function is the expectation value of the quantum mechanical Hamiltonian, as the sum of the kinetic term ( $-h^2 \frac{d^2}{dx^2}$ ) and the polynomial objective function ( $V(x)$ ). The quantum mechanical feature in the system is represented by the set of expectation values $(\phi|x^iu^j|\phi)$ for the possible combination of the kinetic momentum operator ( $u=h\frac{d}{dx}$ ) and the positional operator ( $x$ ). These elements should satisfy the semi-definite condition of a matrix which is the representation of the uncertainty principle. The Plank constant $h$ is a control parameter which would switch from the quantum mechanical case to the classical case, and vice versa. In order to get the optimum, these quantum mechanical expectation values are assumed to be the particle coordinates and propagated by the particle swarm optimization with the restriction. We present a simple example of the computational process.In this article, we present our version of quantum particle swarm optimization. The objective function is the expectation value of the quantum mechanical Hamiltonian, as the sum of the kinetic term ($-h^2 \frac{d^2}{dx^2}$) and the polynomial objective function ($V(x)$). The quantum mechanical feature in the system is represented by the set of expectation values $(\phi|x^iu^j|\phi)$ for the possible combination of the kinetic momentum operator ($u=h\frac{d}{dx}$) and the positional operator ($x$). These elements should satisfy the semi-definite condition of a matrix which is the representation of the uncertainty principle. The Plank constant $h$ is a control parameter which would switch from the quantum mechanical case to the classical case, and vice versa. In order to get the optimum, these quantum mechanical expectation values are assumed to be the particle coordinates and propagated by the particle swarm optimization with the restriction. We present a simple example of the computational process.

Nov 12, 20181 citation

\hbar)

\hbar=1

observables by a linear canonical transformations which is determined by the second order pieces of the quantum mechanical Hamiltonian.

\\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.

h_0

. 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.

For a quantum observable

A_\hbar

depending on a parameter

\hbar

we define the notion ``

A_\hbar

converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that

A_\hbar\to0

is equivalent with

\Vert A_\hbar\Vert\to0

. The

\hbar

-wise product of convergent observables converges to the product of the limiting phase space functions.

\hbar^{-1}

times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the

\hbar

-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.

In this article, we present our version of quantum particle swarm optimization. The objective function is the expectation value of the quantum mechanical Hamiltonian, as the sum of the kinetic term (

-h^2 \frac{d^2}{dx^2}

) and the polynomial objective function (

V(x)

). The quantum mechanical feature in the system is represented by the set of expectation values

(\phi|x^iu^j|\phi)

for the possible combination of the kinetic momentum operator (

u=h\frac{d}{dx}

) and the positional operator (

x

). These elements should satisfy the semi-definite condition of a matrix which is the representation of the uncertainty principle. The Plank constant

h

is a control parameter which would switch from the quantum mechanical case to the classical case, and vice versa. In order to get the optimum, these quantum mechanical expectation values are assumed to be the particle coordinates and propagated by the particle swarm optimization with the restriction. We present a simple example of the computational process.