Coherent state wavefunction

Here, p stand for the final coordinate and momentum of a classical trajectory with initial conditions Xq,Po and the coherent state wavefunction is defined... 

Davies16) has given a rigorous mathematical analysis of the variational estimate of the ground state of the Hamiltonian K based on trial functions consisting of the tensor product x of a molecular wavefunction 0, and a coherent state 0 for the boson field, i.e. one searches for the minimum value of... 

In order to make the END formalism practical, some approximations must be made in the representation of the waveffinctions of the electrons and nuclei. The particulars are outlined in the next section. In the simple model employed in this work, we choose to represent the electronic wavefunction by a group theoretical coherent states parametrization of a single determinant (SD). The nuclear wave function is formulated in terms of a frozen Gaussian wave packet (FGWP) and the limit of a narrow width is taken. This latter approximation corresponds to the classical limit for describing the nuclei. The coherent state representation of a single determinant leads to the so-called Thouless parametrization (29). For the description of the nuclear wavefunction, a quantum description has also been worked out (26, 30), but is yet to be implemented. 

The research group of Th.W. Hansch succeeded in 2002 to trap for the first time cold atoms in such an optical lattice. They cooled at first the atoms below the critical temperature fort BE condensation. Then the well depth of the optical lattice was increased. This transferred the coherent state of the atoms in the free BEC, where all atoms are in the same state i.e. described by the same wavefunction and are therefore not distinguishable, into the incoherent Mott state, where each atom sits on its separate location and can be therefore distinguished from the other atoms. Decreasing the well depth brings the atoms again back into the coherent BEC state. Just by changing the well depth of the optical lattice switches the atomic ensemble from a coherent into an incoherent state and back [1204]. 

The possibility of controlling chemical reactions by selective excitation of the reactants and by coherent control of the excited state wavefunction (as illustrated in Sect. 10.1.4) may open a new fascinating field of laser-induced chemistry. 

It is thus not at all clear to precisely which question the clamped nuclei Hamiltonian provides the answer and a further discussion of the properties of the Coulomb Hamiltonian is required before the clamped nuclei problem can be put into an appropriate form for yielding a potential, There are two main ways in which such a discussion can be attempted. If it is desired to stay with the Coulomb Hamiltonian in its lab oratory-fixed form then the solutions must be expressed in coherent state (wave-packet) form to allow for their continuum nature. If the solutions are required as L -normalizable wavefunctions, then the translational motion must be separated from the Coulomb Hamiltonian and the solutions of the remaining translationally invariant part must be sought It is in this second approach that it is easiest to make contact with the standard arguments and this will be considered in the following section. 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

Considering any of these paradigms, a minimal goal for toy models would be to manipulate the quantum dynamics of a small number of spin levels , and that requires a known and controlled composition of the wavefunction, sufficient isolation and a method for coherent manipulation. As illustrated in Figure 2.13, the first few magnetic states of the system are labelled and thus assigned qubit values. The rest of the spectrum is outside of the computational basis, so one needs to ensure that these levels are not populated during the coherent manipulation. 

Figure 2. Franck-Condon windows lVpc(Gi, r, v5) for the Na3(X) - N83(B) and for the Na3(B) Na3+ (X) + e transitions, X = 621 nm. The FC windows are evaluated as rather small areas of the lobes of vibrational wavefunctions that are transferred from one electronic state to the other. The vertical arrows indicate these regions in statu nascendi subsequently, the nascent lobes of the wavepackets move coherently to other domains of the potential-energy surfaces, yielding, e.g., the situation at t = 653 fs, which is illustrated in the figure. The snapshots of three-dimensional (3d) ab initio densities are superimposed on equicontours of the ab initio potential-energy surfaces of Na3(X), Na3(B), and Na3+ (X), adapted from Ref. 5 and projected in the pseudorotational coordinate space Qx r cos

When the fine structure frequencies fall below 100 MHz they can also be measured by quantum beat spectroscopy. The basic principle of quantum beat spectroscopy is straightforward. Using a polarized pulsed laser, a coherent superposition of the two fine structure states is excited in a time short compared to the inverse of the fine structure interval. After excitation, the wavefunctions of the two fine structure levels evolve at different rates due to their different energies. For example if the nd3/2 and nd5/2 mf = 3/2 states are coherently excited from the 3p3/2 state at time t = 0, the nd wavefunction at a later time t can be written as40... 

The surface-hopping trajectories obtained in the adiabatic representation of the QCLE contain nonadiabatic transitions between potential surfaces including both single adiabatic potential surfaces and the mean of two adiabatic surfaces. This picture is qualitatively different from surface-hopping schemes [2,56] which make the ansatz that classical coordinates follow some trajectory, R(t), while the quantum subsystem wave function, expanded in the adiabatic basis, is evolved according to the time dependent Schrodinger equation. The potential surfaces that the classical trajectories evolve along correspond to one of the adiabatic surfaces used in the expansion of the subsystem wavefunction, while the subsystem evolution is carried out coherently and may develop into linear combinations of these states. In such schemes, the environment does not experience the force associated with the true quantum state of the subsystem and decoherence by the environment is not automatically taken into account. Nonetheless, these methods have provided com-... 

Here q denotes the electronic coordinates, and Q and Q the full sets of nuclear normal coordinates in the photo-excited (a) and ground (g) electronic states. a and Og are the electronic wavefunctions. 0aiy and 0gv are the full nuclear wavefunctions of the two different electronic states, each of which can be decomposed into products of single-mode wavefunctions % and Xgvv/, respectively. Initially the coherence is created by the pumping process, i.e.,... 

The ratio Vo/B determines the transition from coherent diffusive propagation of wavefunctions (delocalized states) to the trapping of wavefunctions in random potential fluctuations (localized states). If I > Vo, then the electronic states are extended with large mean free path. By tuning the ratio Vq/B, it is possible to have a continuous transition from extended to localized states in 3D systems, with a critical value for Vq/B. Above this critical value, wave-functions fall off exponentially from site to site and the delocalized states cannot exist any more in the system. The states in band tails are the first to get localized, since these rapidly lose the ability for resonant tunnel transport as the randomness of the disorder potential increases. If Vq/B is just below the critical value, then delocalized states at the band center and localized states in the band tails could coexist. 

The Cooper pairs are bosons, and below a critical Tc (which is affected by both applied pressure and by applied magnetic field) can condense to the same momentum state and wavefunction for all Cooper pairs in the solid these pairs have long-distance phase coherence and are present in all known superconductors. However, the condensation of these Cooper pair bosons is attributed to electron-phonon coupling only for monoatomic and diatomic metals (BCS theory), where the critical temperature Tc depends on isotopic mass. 

Both the absorption and the resonant tunneling experiments find quantization effects for layer thicknesses of 50 A or less. It is, however, not immediately obvious why the quantum states should be observed even in these thin layers. The discussion of the transport in Chapter 7 concludes that the inelastic mean free path length is about 10-15 A at the mobility edge. The rapid loss of phase coherence of the wavefunction should prevent the observation of quantum states even in a 50 A well, but there are some factors that may explain the observations. The mean free path increases at energies above the... 

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