The quantum model system is considered for which the classic analog is the known Arnold’s саг map. Due to periodicity conditions for the phase space, quantum states are represented by vectors of finite dimension N, and operators by NxN matrices. The integer parameter N characterizes a relative value of quantum effects; classic limit corresponds to N—>∞. Operator тар is suggested which governs discrete time evolution in Heisenberg representation for operators of finite shifts for position and momentum. Explicit form of evolution operator is stated in Schrodinger representation. Solution for non—stationary problem is presented and discussed for initial conditions taken as localized state, two delta—spikes, Gaussian wave packet. Quantum dynamics in terms of Husimi distribution and Wigner function, quasi—cnergy spectrum and eigenvector structure are discussed on а basis of dynamics of Heisenberg operators.
The quantum model system is considered for which the classic analog is the known Arnold’s саг map. Due to periodicity conditions for the phase space, quantum states are represented by vectors of finite dimension N, and operators by NxN matrices. The integer parameter N characterizes a relative value of quantum effects; classic limit corresponds to N—>∞. Operator тар is suggested which governs discrete time evolution in Heisenberg representation for operators of finite shifts for position and momentum. Explicit form of evolution operator is stated in Schrodinger representation. Solution for non—stationary problem is presented and discussed for initial conditions taken as localized state, two delta—spikes, Gaussian wave packet. Quantum dynamics in terms of Husimi distribution and Wigner function, quasi—cnergy spectrum and eigenvector structure are discussed on а basis of dynamics of Heisenberg operators.
The authors initiate the study of ``operator spreading'' in quantum maps on a torus, systems which do not have a tensor product Hilbert space or a notion of spatial locality. The evolution of functions on classical phase space and quantum operator evolutions are compared and contrasted using the perturbed Arnold cat map as an example. Dynamics of operators in quantum chaotic maps are characterized by distinct time scales that can be related to the quasienergy spectrum of the unitary via the spectral form factor.
In this paper we consider some of the consequences of classical stochasticity phenomena in the quantum mechanical Henon–Heiles and Barbanis systems. To explore classical–quantum analogies we have introduced a quantum mechanical phase space based on the coherent state representation. Quantum Poincaré maps (QPMs) were constructed from contour plots of the stationary phase-space densities. We have observed a correlation between the topological features of a QPM at an eigenenergy E and the sensitivity ‖dE/dε‖ of that energy to the strength ε of the nonlinear coupling. States with extreme values of ‖dE/dε‖, corresponding to both high and to low values of ‖dE/dε‖, show regular regions in the quantum phase space, while states with intermediate values of ‖dE/dε‖ span large parts of the quantum phase-space regions. These general features of the QPMs apply for almost each energy multiplet, and no qualitative change in the character of the QPMs was observed above the classical critical energy Ec for the stochastic transition. To investigate the quantum analog of classical trajectories, we have studied the dynamics of initially coherent Gaussian wave packets. We have discovered two limiting types of time evolution of intially coherent wave packets, which exhibit quasiperiodic time evolution or rapid dephasing. Quasiperiodic quantum dynamics is analogous to quasiperiodic classical trajectory dynamics, while rapid dephasing of the wave packet reflects the effects of spreading of the wave packet or/and stochastic classical dynamics. We were able to establish a correspondence between the topology of the quantum phase-space density and the wave packet dynamics. Quasiperiodic time evolution and rapid dephasing are exhibited by wave packets initially located in the regular and in the irregular regions of the quantum phase space, respectively.
The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator AH(t) with Weyl transform AH(t, Q, K), the function A(t,Q)=∫dKAH(t,Q,K) is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum-mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of A(t,Q) can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t0, Q0) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non-Cartesian coordinate systems is given.
In classical mechanics the state of a system is defined by the numerical values of a set of dynamical variables such as the position coordinates and the velocities. In quantum mechanics the state of a system at an instant is represented by a unit vector in a Hilbert space, where a set of axes are the eigenvectors of a complete set of observables. The description of quantum dynamics where the wave function is time-dependent and the operators describing the observables are time-independent is known as the Schrodinger picture. The use of total time derivative of operator that depends on the time evolution of the wave function as well as on any intrinsic time-dependence in the operators simplifies the development of quantum mechanics and allows the development to more closely follow the corresponding development of classical mechanics. The difference between the Schrodinger and Heisenberg pictures is best understood as the difference in the basic set of observables.
The quantum dynamics of minimum uncertainty wave packets in a system described by the surface-state-electron Hamiltonian are studied herein. The quantum evolution is found to be strongly dependent upon the orbit structure in the classical phase space. A Gaussian wave packet, initially centered on the part of the phase space incased by Kolmogorov-Arnold-Moser tori remains localized, while the packets located in the chaotic region spread rapidly, filling the available phase space. To quantify the observed spreading quantum uncertainty is introduced as a dynamical variable, and its time development is determined by the ballistic growth of the position uncertainty and the erratic oscillations of the momentum uncertainty. The quantum evolution is analyzed within the framework of the Floquet formalism. The Floquet spectrum of the stable packet is shown to be dominated by a few quasienergy states with the corresponding Husimi distribution embedded in the classical stability island. Further, the classical concept of diffusion previously used in this context is shown to be inappropriate.
The choice of mathematical representation when describing physical systems is of great consequence, and this choice is usually determined by the properties of the problem at hand. Here we examine the little-known wave operator representation of quantum dynamics and explore its connection to standard methods of quantum dynamics, such as the Wigner phase-space function. This method takes as its central object the square root of the density matrix and consequently enjoys several unusual advantages over standard representations. By combining this with purification techniques imported from quantum information, we are able to obtain a number of results. Not only is this formalism able to provide a natural bridge between phase- and Hilbert-space representations of both quantum and classical dynamics, we also find the wave operator representation leads to semiclassical approximations of both real and imaginary time dynamics, as well as a transparent correspondence to the classical limit. It is then demonstrated that there exist a number of scenarios (such as thermalization) in which the wave operator representation possesses an equivalent unitary evolution, which corresponds to nonlinear real-time dynamics for the density matrix. We argue that the wave operator provides a new perspective that links previously unrelated representations and is a natural candidate model for scenarios (such as hybrids) in which positivity cannot be otherwise guaranteed.