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Quantum Reports

Hamilton Equations, Commutator, and Energy Conservation

Weng Cho Chew, Aiyin Y. Liu, C. Salazar-Lazaro, Dong-Yeop Na, Wei E. I. Sha

Published Dec 9, 2019• 6 citations
Publisher

Abstract

We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations.

DOI
10.3390/quantum1020027

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We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.
Classical and quantum physics represent two distinct theories; however, quantum physics is regarded as the more fundamental of the two. It is posited that classical mechanics should arise from quantum mechanics under certain limiting conditions. Nevertheless, this remains a challenging objective. In this work, we explore the potential for unifying the dynamics of classical and quantum physics. This discussion does not suggest that classical behavior emerges from quantum mechanics; rather, it demonstrates the exact equivalence between the dynamics of quantum observables and their classical counterparts. It is shown that the Heisenberg equations of motion can be cast in a form that is structurally equivalent to Newton's equations of motion, with ℏ\hbarℏ dropping out from the equations. In a generalized analysis, the Heisenberg equations are cast in a form that is identical to the classical Hamilton's equations of motion. This implies that both quantum and classical dynamics are governed by the same form of equations, with the Heisenberg operators substituting the classical observables.
The canonical commutation relation is a fundamental postulate of the quantum theory regarding the operators needed in quantum description of physical observables. It is shown that the Fourier transform is derivable from this seemingly simple postulate along with the basic properties of the position and momentum operators. Further discussions on the canonical commutation relation reveal its connection to a more fundamental notion that energy must be conserved. This discussion also unveils the mathematical homomorphism between the classical and quantum theories for systems represented by sum separable Hamiltonians. Another link between the classical and quantum theories is established by the correspondence principle which states that the classical theory emerges from quantum theory in the limit of vanishingly small Planck constant. Finally, the quantum Maxwell’s equations, which have been derived in our previous works, are presented and briefly discussed, and the 3-D mode transform is derived that can be interpreted as a generalization of the Fourier transform. We present both the details and meanings of the 3-D mode transform which will serve as a foundation for a full 3-D quantum finite-difference time-domain method.
Hamiltonian mechanics describes the evolution of a system through its Hamiltonian. The Hamiltonian typically also represents the energy observable, a Noether-conserved quantity associated with the time invariance of the law of evolution. In both quantum and classical mechanics, Hamiltonian mechanics demands a precise relationship between time evolution and observable energy, albeit using slightly different terminology. We distill basic conditions satisfied in both quantum and classical mechanics, including canonical coordinate symmetries and inner product invariance. We express these conditions in the framework of generalized probabilistic theories, which includes generalizing the definition of energy eigenstates in terms of time-invariant properties of the Hamiltonian system. By postulating these conditions to hold, we derive a unified Hamiltonian system model. This unified framework describes quantum and classical mechanics in a consistent language, facilitating their comparison. We moreover discuss alternative theories: an equation of motion given by a mixture of commutation relations, an information-restricted version of quantum theory, and Spekken's toy theory. The findings give a deeper understanding of the Hamiltonian in quantum and classical theories and point to several potential research topics.
The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.
The classical Landau-Lifshitz equation has been derived from quantum mechanics. Starting point is the assumption of a non-Hermitian Hamilton operator to take the energy dissipation into account. The corresponding quantum mechanical time dependent Schrödinger, Liouville and Heisenberg equation have been described and the similarities and differences between classical and quantum mechanical spin dynamics have been discussed. Furthermore, a time dependent Schrödinger equation corresponding to the classical Landau-Lifshitz-Gilbert equation and two ways to include temperature into the quantum mechanical spin dynamics have been proposed.