Potential time-dependent

Expanding Waves. As a further application we tmn to the expanding potential problem [261-263], where we shall work from the amplitude modulus to the phase. The time-dependent potential is of the form... 

One drawback is that, as a result of the time-dependent potential due to the LHA, the energy is not conserved. Approaches to correct for this approximation, which is valid when the Gaussian wavepacket is narrow with respect to the width of the potential, include that of Coalson and Karplus [149], who use a variational principle to derive the equations of motion. This results in replacing the function values and derivatives at the central point, V, V, and V" in Eq. (41), by values averaged over the wavepacket. 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

In voltammetry a time-dependent potential is applied to an electrochemical cell, and the current flowing through the cell is measured as a function of that potential. A plot of current as a function of applied potential is called a voltammogram and is the electrochemical equivalent of a spectrum in spectroscopy, providing quantitative and qualitative information about the species involved in the oxidation or reduction reaction.The earliest voltammetric technique to be introduced was polarography, which was developed by Jaroslav Heyrovsky... 

If we fix a realization of the path Q(x), then, when performing the path integration over q, the particle may be treated as acted on by a time-dependent potential V-miiQix), q). From traditional quantum mechanics it is clear that this integration is equivalent to the solution of the time-dependent Shrodinger equation in imaginary time. 

An alternative approximation to the adiabatic probability is to invoke an instantaneous equilibrium-like probability. In the context of the work theorem, Hatano and Sasa [72] analyzed a nonequilibrium probability distribution that had no memory, and others have also invoked a nonequilibrium probability distribution that is essentially a Boltzmann factor of the instantaneous value of the time-dependent potential [73, 74]. 

Suppose that the atom (or nucleus) initially in an eigenstate 1 is subjected to a small time-dependent potential V (t) on top of the unperturbed Hamiltonian Ho2 It is then possible to treat the coefficients an in Eq. (A3.12) as functions of time, with ai 2(r) 1 being the probability that it is still in state 1 after a time x and a2 2(t) < C 1 the probability that it has undergone a transition to another eigenstate 2 . Substituting in Schrodinger s equation (A3.8),... 

Time-Independent and Periodic Time-Dependent Potentials.73... 

DENSITY FUNCTIONAL THEORY OF MANY-ELECTRON SYSTEMS FOR TIME-INDEPENDENT AND PERIODIC TIME-DEPENDENT POTENTIALS... 

We denote the fluctuations of the number density of the monomers of component j at a point r and at a time t as pj r,t). With this definition we have pj(r,t))=0. In linear response theory, the Fourier-Laplace transform of the time-dependent mean density response to an external time dependent potential U r,t) is expressed as ... 

Although a variety of electtochemical methods are available, voltammetry (based on the record of the current i flowing across an electrochemical cell under the application of a given time-depending potential E) has become the most important measuring technique in pure and applied elecPochemistry [79]. 

Before continuing the study of the dynamics of the inflationary phase, let us focus on one specific example of inflationary scenario chaotic inflation. Historically, this was not the first model that was proposed but we think it was the first to provide a satisfactory scenario. The main difficulty with inflation is to have the slow roll conditions to be satisfied at some epoch. Indeed, as we saw, one need to put the field away from the minimum of its potential for the inflaton to behave like a cosmological constant. The first inflationary model ( Guth 1981) supposed a potential like that of Eq. (7.28) where the field slowly moved away from its minimum because of a phase transition. However, this led to a number of difficulties, see for example Ref. (Liddle Lyth). Fortunately, it was soon realized that it was not necessary to have a time dependent potential for inflation to proceed. Linde (Linde 1985) noticed that inflation could start as soon as the Universe would exit the Planck era. The idea was that it is reasonable to suppose that at the end of the Planck era (when p > ), no large-scale correlation could be expected in the scalar field, so that one could expect very irregular (hence, chaotic) initial condition with... 

An optical consideration specific to the use of the TOF-MS with high-intensity sources is the removal of background ions, plasma gas ions, and matrix ions to prevent detector saturation. To date, this has been accomplished with parallel-plate deflection in the flight path, which is depicted in Fig. 12.6. Removal of specific m/z ions is accomplished by the application of a time-dependent potential to one, or both, of the plates at some time delayed with respect to the repeller pulse. In this way, those ions between the plates at the time of the pulse experience a field transverse to the flight axis and are removed from the flight path. The voltage pulses employed here must have fast rise and fall times (<10-20 nsec), and be applied at precise delay intervals to ensure that mass resolution is not compromised, and to allow for the unimpeded passage of the previous and subsequent masses. Also, the... 

Third, consider a triangular cyclic potential sweep under reversible (Nerns-tian) conditions for a planar electrode (Fig. 6.10), to derive a cyclic voltam-mogram (Fig. 6.11) In the Nemst expression, Eq. (6.19.14) a time-dependent potential is added E(t) = E(0) — vt, whence... 

The time-dependent potential Vc(t) across a capacitor C is given by the integral ... 

Potentiodynamictechniques— are all those techniques in which a time-dependent -> potential is applied to an - electrode and the current response is measured. They form the largest and most important group of techniques used for fundamental electrochemical studies (see -> electrochemistry), -> corrosion studies, and in -> electroanalysis, -+ battery research, etc. See also the following special potentiodynamic techniques - AC voltammetry, - DC voltammetry, -> cyclic voltammetry, - linear scan voltammetry, -> polarography, -> pulse voltammetry, - reverse pulse voltammetry, -> differential pulse voltammetry, -> potentiodynamic electrochemical impedance spectroscopy, Jaradaic rectification voltammetry, - square-wave voltammetry. 

Time-dependent density functional theory (TDDFT) as a complete formalism [7] is a more recent development, although the historical roots date back to the time-dependent Thomas-Fermi model proposed by Bloch [8] as early as 1933. The first and rather successful steps towards a time-dependent Kohn-Sham (TDKS) scheme were taken by Peuckert [9] and by Zangwill and Soven [10]. These authors treated the linear density response of rare-gas atoms to a time-dependent external potential as the response of non-interacting electrons to an effective time-dependent potential. In analogy to stationary KS theory, this effective potential was assumed to contain an exchange-correlation (xc) part, r,c(r, t), in addition to the time-dependent external and Hartree terms ... 

One-to-One Mapping Between Time-Dependent Potentials and Time-Dependent Densities... 

Here K(r) is the one-particle potential in which the electron moves and VA — r ) is the bare Coulomb potential between the electrons. Fe(rt) is the external space-and time-dependent potential which acts as a source coupled to the electron density operator, a is the spin index, and is such that... 

The application of a time-var3ung potential diflTerence to a parallel-plate capadtor must lead to a time-dependent potential gradient, — E(t). In view of the relation D — egE + P [equation (19) ] both D and P must be functions of time. Just how D and P vary with time will depend upon the dynamic properties of the dielectric slab. In the absence of a dielectric (P = 0), of course, Do(0 = 61 (0. and hence the ratio... 

The interaction between the atoms is represented by a time-dependent potential W. This leads to a time-dependent Schrodinger equation describing the entire weakly coupled system ... 

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