Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Heisenberg Operator Approach: Nano-Mechanical Oscillator (Part 2) and the Quantum LC Circuit

Duncan G. Steel

Published Apr 30, 2021• 0 citations
Publisher

Abstract

Abstract With the knowledge of the new design rules in Chapter 7, we use this new insight to find the eigenvectors for the nano-vibrator problem, and then we use the same approach to examine the quantum LC circuit. While the usual approach is to use Kirchhoff’s laws to analyze a simple circuit classically, we first see that Hamilton’s equations can in fact be used, giving the same classical result. But then, using the new design rules and the knowledge of the total energy in the circuit, we identify a canonical coordinate and a conjugate momentum that have nothing to do with real space and motion of a particle of mass m. At the same time, consistent with the Schrödinger picture, we continue to see that the time evolution of an observable such as position, x(t), or current, i(t), is not part of the solution. Given that Hamilton’s equations give the same result as Kirchhoff’s law but the quantum solution does not, reinforces the idea that the quantum description is showing features that cannot be imagined with a viewpoint based on classical (i.e. non-quantum) analysis.

DOI
10.1093/oso/9780192895073.003.0008

Related papers

Nano-Mechanical Oscillator and Basic Dynamics: Part I

Duncan G. Steel

Abstract This discussion introduces the student to the reality, in quantum technology, that analysis of any problem necessarily begins with the Hamiltonian representing the system. The quantum Hamiltonian represents the total energy of the system, the sum of kinetic energy plus potential energy, written in canonical coordinates and conjugate momenta, and where these variables become time independent quantum operators. The nature of the potential energy for the nano-vibrator, following Hooke’s law, serves to localize the particle. The relevance of the nano-vibrator Hamiltonian—sometimes called the harmonic oscillator Hamiltonian—is perhaps one of the most important Hamiltonians in quantum systems. Not only can it be extended to cover things like phonons in solids, vibrations in molecules, and the behavior of bosons, but it is also the basis for leading to the concept of a photon, the quantum radiation field, and the quantum vacuum. This chapter provides the basic introduction for vibration of a particle or a nano-rod and looks at the wave-like behavior that emerges from the solution to the time independent Schrödinger equation. When we include the time evolution, we can observe dynamical behavior and begin to examine the meaning of quantum measurement.

•Apr 30, 2021•0 citations

Quantum mechanics from Newton's second law and the canonical commutation relation [ X , P ] = i

Mark C. Palenik

Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations $F=\frac{dP}{dt}$, $P=m\frac{dV}{dt}$, and $\left[X,P\right]=i$. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.

arXiv:1402.0396•European Journal of Physics•May 9, 2014•4 citations

q-Equivalent Hamiltonian Operators

Mohsen Razavy

From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.

Canadian Journal of Physics•Sep 1, 1972•19 citations

Algebra of quantum mechanics via classical phonons. II: Klein-Gordon dynamics, the Heisenberg formalism, the Dirac canonical commutation rule and the Poincare algebra through the continuous Poisson bracket formalism

Emmanuel Giner

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.

quant-pharXiv:2607.03909•Jul 7, 2026•0 citations

Weyl quantization

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.

Cambridge University Press eBooks•Mar 11, 2004•0 citations

Quantum Synchronization in Nonconservative Electrical Circuits with Kirchhoff-Heisenberg Equations

M. Mariantoni, Noah Gorgichuk

We investigate quantum synchronization phenomena in electrical circuits that incorporate specifically designed nonconservative elements. A dissipative theory of classical and quantized electrical circuits is developed based on the Rayleigh dissipation function. The introduction of this framework enables the formulation of a generalized version of classical Poisson brackets, which are termed Poisson-Rayleigh brackets. By using these brackets, we are able to derive the equations of motion for a given circuit. Remarkably, these equations are found to correspond to Kirchhoff's current laws when Kirchhoff's voltage laws are employed to impose topological constraints, and vice versa. In the quantum setting, the equations of motion are referred to as the Kirchhoff-Heisenberg equations, as they represent Kirchhoff's laws within the Heisenberg picture. These Kirchhoff-Heisenberg equations, serving as the native equations for an electrical circuit, can be used in place of the more abstract master equations in Lindblad form. To validate our theoretical framework, we examine three distinct circuits. The first circuit consists of two resonators coupled via a nonconservative element. The second circuit extends the first to incorporate weakly nonlinear resonators, such as transmons. Lastly, we investigate a circuit involving two resonators connected through an inductor in series with a resistor. This last circuit, which incidentally represents a realistic implementation, allows for the study of a singular system, where the absence of a coordinate leads to an ill-defined system of Hamilton's equations. To analyze such a pathological circuit, we introduce the concept of auxiliary circuit element. After resolving the singularity, we demonstrate that this element can be effectively eliminated at the conclusion of the analysis, recuperating the original circuit.

arXiv:2403.10474•Mar 15, 2024•1 citation
Abstract With the knowledge of the new design rules in Chapter 7, we use this new insight to find the eigenvectors for the nano-vibrator problem, and then we use the same approach to examine the quantum LC circuit. While the usual approach is to use Kirchhoff’s laws to analyze a simple circuit classically, we first see that Hamilton’s equations can in fact be used, giving the same classical result. But then, using the new design rules and the knowledge of the total energy in the circuit, we identify a canonical coordinate and a conjugate momentum that have nothing to do with real space and motion of a particle of mass m. At the same time, consistent with the Schrödinger picture, we continue to see that the time evolution of an observable such as position, x(t), or current, i(t), is not part of the solution. Given that Hamilton’s equations give the same result as Kirchhoff’s law but the quantum solution does not, reinforces the idea that the quantum description is showing features that cannot be imagined with a viewpoint based on classical (i.e. non-quantum) analysis.
Abstract This discussion introduces the student to the reality, in quantum technology, that analysis of any problem necessarily begins with the Hamiltonian representing the system. The quantum Hamiltonian represents the total energy of the system, the sum of kinetic energy plus potential energy, written in canonical coordinates and conjugate momenta, and where these variables become time independent quantum operators. The nature of the potential energy for the nano-vibrator, following Hooke’s law, serves to localize the particle. The relevance of the nano-vibrator Hamiltonian—sometimes called the harmonic oscillator Hamiltonian—is perhaps one of the most important Hamiltonians in quantum systems. Not only can it be extended to cover things like phonons in solids, vibrations in molecules, and the behavior of bosons, but it is also the basis for leading to the concept of a photon, the quantum radiation field, and the quantum vacuum. This chapter provides the basic introduction for vibration of a particle or a nano-rod and looks at the wave-like behavior that emerges from the solution to the time independent Schrödinger equation. When we include the time evolution, we can observe dynamical behavior and begin to examine the meaning of quantum measurement.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.
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.
The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.
We investigate quantum synchronization phenomena in electrical circuits that incorporate specifically designed nonconservative elements. A dissipative theory of classical and quantized electrical circuits is developed based on the Rayleigh dissipation function. The introduction of this framework enables the formulation of a generalized version of classical Poisson brackets, which are termed Poisson-Rayleigh brackets. By using these brackets, we are able to derive the equations of motion for a given circuit. Remarkably, these equations are found to correspond to Kirchhoff's current laws when Kirchhoff's voltage laws are employed to impose topological constraints, and vice versa. In the quantum setting, the equations of motion are referred to as the Kirchhoff-Heisenberg equations, as they represent Kirchhoff's laws within the Heisenberg picture. These Kirchhoff-Heisenberg equations, serving as the native equations for an electrical circuit, can be used in place of the more abstract master equations in Lindblad form. To validate our theoretical framework, we examine three distinct circuits. The first circuit consists of two resonators coupled via a nonconservative element. The second circuit extends the first to incorporate weakly nonlinear resonators, such as transmons. Lastly, we investigate a circuit involving two resonators connected through an inductor in series with a resistor. This last circuit, which incidentally represents a realistic implementation, allows for the study of a singular system, where the absence of a coordinate leads to an ill-defined system of Hamilton's equations. To analyze such a pathological circuit, we introduce the concept of auxiliary circuit element. After resolving the singularity, we demonstrate that this element can be effectively eliminated at the conclusion of the analysis, recuperating the original circuit.