We propose a general framework of the combined description of a classical system with a quantum system by representing the quantum system as a Hamiltonian field and defined the evolution of the whole system as Hamiltonian flow. We illustrate the general formalism in two cases, namely the classical particle coupled with a quantum oscillator and classical particle with quantum spin in electromagnetic field.
We propose a general framework of the combined description of a classical system with a quantum system by representing the quantum system as a Hamiltonian field and defined the evolution of the whole system as Hamiltonian flow. We illustrate the general formalism in two cases, namely the classical particle coupled with a quantum oscillator and classical particle with quantum spin in electromagnetic field.
We describe quantum and classical Hamiltonian dynamics in a common Hilbert space framework, that allows the treatment of mixed quantum-classical systems. The analysis of some examples illustrates the possibility of entanglement between classical and quantum systems. We give a summary of the main tools of Berezin-Toeplitz and geometric quantization, that provide a relation between the classical and the quantum models, based essentially on the selection of a subspace of the classical Hilbert space. Coherent states provide a systematic tool for the inverse process, called dequantization, that associates a classical Hamiltonian system to a given quantum dynamics through the choice of a complete set of coherent states.
An extension of Hamiltonian dynamics, defined on hyper-Kahler manifolds ('hyper-Hamiltonian dynamics') and sharing many of the attractive features of standard Hamiltonian dynamics, was introduced in previous work. In this paper, we discuss applications of the theory to physically interesting cases, dealing with the dynamics of particles with spin 1/2 in a magnetic field, i.e. the Pauli and the Dirac equations. While the free Pauli equation corresponds to a hyper-Hamiltonian flow, it turns out that the hyper-Hamiltonian description of the Dirac equation, and of the full Pauli one, is in terms of two commuting hyper-Hamiltonian flows. In this framework one can use a factorization principle discussed here (which is a special case of a general phenomenon studied by Walcher) and provide an explicit description of the resulting flow. On the other hand, by applying the familiar Foldy?Wouthuysen and Cini?Tousheck transformations (and the one recently introduced by Mulligan) which separate?in suitable limits?the Dirac equation into two equations, each of these turn out to be described by a single hyper-Hamiltonian flow. Thus the hyper-Hamiltonian construction is able to describe the fundamental dynamics for particles with spin.
The formalism of quantum mechanics is presented in a way that its interpretation as a classical field theory is emphasized. Two coupled real fields are defined with given equations of motion. Densities and currents associated to the fields are found with their corresponding conserved quantities. The behavior of these quantities under a galilean transformation suggest the association of the fields with a quantum mechanical free particle. An external potential is introduced in the Lagrange formalism. The description is equivalent to the conventional Schrödinger equation treatment of a particle. We discuss the attempts to build an interpretation of quantum mechanics based on this scheme. The fields become the primary onthology of the theory and the particles appear as emergent properties of the fields. These interpretations face serious problems for systems with many degrees of freedom.
Based on the Symmetry-Adapted Product Operator (SPO), the Moltiple Quantum Coherence Product Operator (MQCPO) formalism is introduced in this paper. The modified density operator theory is universal for the description of modern multiple pulse and two-dimensional NMR experiments performed on InS(I = 1/2, 5 = 1/2; n = an arbitray integer number) spin systems. In fact, MQCPO is a Hermitian combination of irredusible tensor operators for a compsite particle, and reveals symmetry properties of the spin system in a different aspect from SAPO. There exist simple linear relations between MQCPO and SAPO for In(I = 1/2, n = 2,3) spin systems, the SAPO forms of MQCPO are listed as a usual example. MQCPO shows a simple cyclic commutation relationship with the free procession Hamiltonian, but not with the Hamiltonian of a pulse effect. Therefore, MQCPO is convenient for description of evolution period. Using the linear relations between MQCPO and SAPO, as well as the permutation properties for the coordinate indices of Cartesian components in a SAPO, the description of a pulse action tarns out to be easy and direct under the "z"-representation. Analyses of the pulse sequences for heteronuclear spectra editing and spin topology filtration have shown the power of this formalism.
Quantum-classical mixing is studied by a group-theoretical approach, and a quantum-classical equation of motion is derived. The quantum-classical bracket entering the equation preserves the Lie algebra structure of quantum and classical mechanics, and, therefore, leads to a natural description of interaction between quantum and classical degrees of freedom. The exact formalism is applied to coupled quantum and classical oscillators. Various approximations, such as the mean-field and the multiconfiguration mean-field approaches, which are of great utility in studying realistic multidimensional systems, are derived. Based on the formulation, a natural classification of the previously suggested quantum-classical equations of motion arises, and several problems from earlier works are resolved.
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.