Heisenberg equations dynamics

As long as the system can be described by the rate constant - this rules out the localization as well as the coherent tunneling case - it can with a reasonable accuracy be considered in the imaginary-time framework. For this reason we rely on the Im F approach in the main part of this section. In a separate subsection the TLS real-time dynamics is analyzed, however on a simpler but less rigorous basis of the Heisenberg equations of motion. A systematic and exhaustive discussion of this problem may be found in the review [Leggett et al. 1987]. 

The wave function is an irreducible entity completely defined by the Schrbdinger equation and this should be the cote of the message conveyed to students. It is not useful to introduce any hidden variables, not even Feynman paths. The wave function is an element of a well defined state space, which is neither a classical particle, nor a classical field. Its nature is fully and accurately defined by studying how it evolves and interacts and this is the only way that it can be completely and correctly understood. The evolution and interaction is accurately described by the Schrbdinger equation or the Heisenberg equation or the Feynman propagator or any other representation of the dynamical equation. 

The actual form of Heisenberg s dynamic equation can be constructed when the expression... 

A similar calculation may be carried out for the time evolution of an observable. Starting with the Heisenberg equation of motion for a dynamical variable B,... 

There are several theoretical approaches that can be used to calculate the dynamics and correlation properties of two atoms interacting with the quantized electromagnetic held. One of the methods is the wavefunction approach in which the dynamics are given in terms of the probability amplitudes [9]. Another approach is the Heisenberg equation method, in which equations of motion for the atomic and held operators are found from the Hamiltonian of a given system [10], The most popular approach is the master equation method, in which the equation of motion is found for the density operator of an atomic system weakly coupled to a system regarded as a reservoir [7,8,41], There are many possible realizations of reservoirs. The typical reservoir to which atomic systems are coupled is the quantized three-dimensional multimode vacuum held. The major advantage of the master equation is that it allows us to consider the evolution of the atoms plus held system entirely in terms of atomic operators. 

Connected to our derivation of the Heisenberg equation of motion is Ehren-fest s theorem for the time evolution of expectation values. We leave aside all explicit dependences of the state function and of the operator on dynamical variables and thereby make no reference to the particular choice of picture. If we consider the expectation value of an observable O for a normalized state Y, Eq. (4.25), then its time derivative is given by... 

In Schrodinger s wave mechanics (which has been shown4 to be mathematically identical with Heisenberg s quantum mechanics), a conservative Newtonian dynamical system is represented by a wave function or amplitude function [/, which satisfies the partial differential equation... 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

However, the Heisenberg group formalism is a very useful tool to represent quantum and classical dynamical quantities, such as observables and equations of motion, only when a prescription on the generator of the time evolution exists. The comparison with the fully quantum or fully classical dynamics allows us to deduce only the formal properties that the mixed quantum-classical brackets have to satisfy in order to generate a consistent evolution, but does... 

The heavy superstructure of modern quantum mechanics rests largely upon a set of mathematical relationships published in 1927 by Heisenberg. These relationships are now usually referred to collectively as the uncertainty principle. Heisenberg showed that in any quantum-mechanical system, pairs of dynamical variables for particles can be simultaneously and sharply defined only if their operators commute. This means only if their operators H and K satisfy the equation... 

In the standard theory of quantum mechanics, two kinds of evolution processes are introduced, which are qualitatively different from each other. One is the spontaneous process, which is a reactive (unitary) dynamical process and is described by the Heisenberg or Schrodinger equation in an equivalent manner. The other is the measurement process, which is irreversible and described by the von Neumann projection postulate [26], which is the rigorous mathematical form of the reduction of the wave packet principle. The former process is deterministic and is uniquely described, while the latter process is essentially probabilistic and implies the statistical nature of quantum mechanics. 

We now introduce the ideas of Weyl to distinguish between pure states and mixtures. Pure states were mathematically represented by eigenvectors of observables, which described the properties of a particle or a dynamic state. On the other hand, mixtures were composed of pure states of a certain mixing relationship. These aspects are clearly important to chemists and obviously to the electrochemists too. The canonical variables, G and H [19], have to satisfy the canonical or Heisenberg commutation relation, derived from Equations 3.12 and 3.13 ... 

The matter may be regarded from the point of view of the uncertainty principle. The behaviour of particles which is defined by the wave equation is equivalent to an indefiniteness in what may be known of their dynamical coordinates, lip and q are the momentum and position coordinates, Heisenberg s principle states that both cannot be known simultaneously except with a range of uncertainty given by the relation ApAg = h, approximately. In the temperature... 

Finally, the correctness of all this equivalent construction and commutators may be proved by considering the dynamical equation of a given operator in the Heisenberg picture it becomes successively ... 

The time-independent Schrodinger equation (SE) for a molecular system derives from Hamiltonian classical dynamics and includes atomic nuclei as well as electrons. Eigenfunctions are therefore functions of both electronic and nuclear coordinates. Very often, however, the nuclear and electronic variables can be separated. The motion of the heavy particles may be treated using classical mechanics. Particularly at high temperatures, the Heisenberg uncertainty relation Ap Ax > /i/2 is easy to satisfy for atomic nuclei, which have a particle mass at least 1836 times the electron mass. The immediate problem for us is to obtain a time-independent SE including not only the electrons but also the nuclei and subsequently solve the separation problem. 

The time evolution of the quantum dynamical variable can be generated by using Eq. (233). This result is the formal solution of Heisenberg s equation of motion... 

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