Isolated quantum states

The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

Chemical reaction dynamics is an attempt to understand chemical reactions at tire level of individual quantum states. Much work has been done on isolated molecules in molecular beams, but it is unlikely tliat tliis infonnation can be used to understand condensed phase chemistry at tire same level [8]. In a batli, tire reacting solute s potential energy surface is altered by botli dynamic and static effects. The static effect is characterized by a potential of mean force. The dynamical effects are characterized by tire force-correlation fimction or tire frequency-dependent friction [8]. 

Much of the regularity in classical systems can often be best discerned directly by observing their spatial power spectra (see section 6.3). We recall that in the simplest cases, the spectra consist of few isolated discrete peaks in more complex chaotic evolutions, we might get white noise patterns (such as for elementary additive rules). A discrete fourier transform (/ ) of a typical quantum state is defined in the most straightforward manner ... 

So far we considered only instantaneous quantum states. The way a wave function evolves in time is given (for a closed isolated system) by the Schrod-inger equation... 

To describe a chemical reaction from a physical standpoint at the nonrelati-vistic level, one must first construct the Hilbert space associated with all quantum states related to the system defined by its molecular hamiltonian, Hm. For the isolated system the time-dependent Schrodinger equation... 

The outcome of an isolated (microscopic) reactive scattering event can be specified in terms of an intrinsic fundamental quantity the reaction cross-section. The cross-section is an effective area that the reactants present to each other in the scattering process. It depends on the quantum states of the molecules as well as the relative speed of the reactants, and it can be calculated from the collision dynamics (to be described in Chapter 4). 

Abstract Interaction between a quantum system and its surroundings - be it another similar quantum system, a thermal reservoir, or a measurement device - breaks down the standard unitary evolution of the system alone and introduces open quantum system behaviour. Coupling to a fast-relaxing thermal reservoir is known to lead to an exponential decay of the quantum state, a process described by a Lindblad-type master equation. In modern quantum physics, however, near isolation of individual quantum objects, such as qubits, atoms, or ions, sometimes allow them only to interact with a slowly-relaxing near-environment, and the consequent decay of the atomic quantum state may become nonexponential and possibly even nonmonotonic. Here we consider different descriptions of non-Markovian evolutions and also hazards associated with them, as well as some physical situations in which the environment of a quantum system induces non-Markovian phenomena. 

We use the same near-field interferometer as in the previous section and introduce various gases at low, controllable pressure (between 5.10-8 mbar and 2.5 x 10-6 mbar) into the vacuum chamber. Each collision between a fullerene molecule and a gas particle entangles their motional states. In order to obtain the properties of the isolated quantum system one has to trace over the state of the scattered molecule. We assume that the mass of the fullerene molecule is much greater than the mass mg of the gas particle and find that the density operator for the fullerene molecule alone, po(r f1), is modified by a multiplicative factor because of the collision... 

In brief, we propose the TOT electro-optical setup in which electrical measurements have a high quality factor combined with the coherence of an all optical experiment. The TOT can be easily incorporated in electrical circuits as a nonlinear element ensuring a scalability of the architecture. The quantum dot isolation in the TOT will protect entanglement of quantum states thus permitting field programmable gates arrays. 

Here, a state 0) is perturbed by an external field represented by the perturbation operator V(f) that describes the interactions between the quantum subsystem and the external field. The Hamiltonian Hq describes the isolated quantum mechanical subsystem and denotes the interaction operator describing the interactions... 

A.IO.10 The ergodic hypothesis slates that a single isolated system spends equal time in each of the available quantum states. 

Let A and B be systems pertaining to our ensemble. We may then ask what is the probability of finding A in a quantum state characterized by whereas B is simultaneously in a quantum state represented by an energy eigenvalue Ej( u,b) As we tacitly assumed the interactions between systems in the ensemble to be very weak, so that the spectrum of energy eigenstates is essentially that of an isolated system, we conclude that the answer to the question is given by... 

This represents a formidable practical problem, as one is very unlikely to find isolated atoms with two nonorthogonal dipole moments and quantum states close in energy. Consider, for example, a V-type atom with the upper states 11), 3) and the ground state 2). The evaluation of the dipole matrix elements produces the following selection rules in terms of the angular momentum quantum numbers J — J2 = 1,0, J3 — J2 = 1,0, and Mi — M2 = M3 — M2 = 1,0. Since Mi / M3, in many atomic systems, p12 is perpendicular to p32 and the atomic transitions are independent. Xia et al. [62] have found transitions with parallel and antiparallel dipole moments in sodium molecules (dimers) and have demonstrated experimentally the effect of quantum interference on the fluorescence intensity. We discuss the experiment in more details in the next section. Here, we point out that the transitions with parallel and antiparallel dipole moments in the sodium dimers result from a mixing of the molecular states due to the spin-orbit coupling. 

Fig. 1. Example of a hexa-ferf-butyldecacyclene (HBDC) molecular rotor, which is supposed to rotate around the X-X axis. Isolated in the gas phase, the quantum state of this molecule can be prepared in a rotational wave packet to start the described rotation before quantum dilution of the wave packet...

Let us consider the preparation of an isolated quantum system in a non-stationary state where a wave packet initiates a permanent quantum motion (a coherent process) in this system. This situation is similar to, for example, the... 

From statistical mechanics the second law as a general statement of the inevitable approach to equilibrium in an isolated system appears next to impossible to obtain. There are so many different kinds of systems one might imagine, and each one needs to be treated differently by an extremely complicated nonequilibrium theory. The final equilibrium relations however involving the entropy are straightforward to obtain. This is not done from the microcanonical ensemble, which is virtually impossible to work with. Instead, the system is placed in thermal equilibrium with a heat bath at temperature T and represented by a canonical ensemble. The presence of the heat bath introduces the property of temperature, which is tricky in a microscopic discipline, and relaxes the restriction that all quantum states the system could be in must have the same energy. Fluctuations in energy become possible when a heat bath is connected to the... 

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