Schrodinger evolution equation

We have arrived at the quantum master equation (5.6) heuristically via the Schrodinger-Langevin equation (5.3). It turns out, however, that (5.6) has a firmer foundation it can be proved mathematically on the basis of the following three general conditions concerning the evolution of p. 

Time reversal in quantum systems also leads to different results compared to classical systems. The basic relationship between the Hamiltonian (or energy) operator and time evolution in quantum mechanics is defined by Schrodinger s equation as... 

The Wigner function has the valuable property that the time evolution equation for the quantum dynamics in the Wigner representation resembles that for the classical Liouville dynamics. Specifically, the Schrodinger equation can be transformed to [70]... 

From the general evolution equation (Equation (34)) of the Hermitian operator, the equation of motion of the density operator in Schrodinger picture could be derived as follows ... 

Note that the invariance of quantum observables under unitary transformations has enabled us to represent quantum time evolutions either as an evolution of the wavefunction with the operator fixed, or as an evolution of the operator with constant wavefunctions. Equation (2.1) describes the time evolution of wavefunctions in the Schrodinger picture. In the Heisenberg picture the wavefunctions do not evolve in time. Instead we have a time evolution equation for the Heisenberg operators ... 

It is actually simple to find a formal time evolution equation in P space. This formal simplicity stems from the fact that the fundamental equations of quantum dynamics, the time-dependent Schrodinger equation or the Liouville equation, are linear. Starting from the quantum Liouville equation (10.8) forthe overall system— system and bath. 

We prefer to write the time-dependent free Dirac equation as a quantum-mechanical evolution equation (that is, in the familiar Schrodinger form ) in the following way... 

We note that the choice of a Hilbert space of square-integrable functions as the state space of the evolution equation is perfectly natural for the Schrodinger equation. The solutions of the Schrodinger equation are in the Hilbert space L (R ) (they have only one component), and the expression tj x,t) is interpreted as a density for the position probability at time t. Hence the norm of a Schrodinger wave packet,... 

Now, the Hamiltonian canonic field equations provide the direct Schrodinger equation when the field evolution equation is considered. 

With this appears the question of the U-transformation of the unitary operator in Schrodinger picture (3.310) for doing this one has to employ the evolution equations in Schrodinger- and U-pictures, respectively as ... 

Postulate III gives the time evolution equation for the wave function iff (time-dependent Schrodinger equation Ht / = ih ), using the energy operator Hamiltonian H). 

What would happen if one prepared the system in a given state which does not represent a stationary state For example, one may deform a molecule by using an electric field and then switch the field off. The molecule will suddenly turn out to be in state ip, that is not its stationary state. Then, according to quantum mechanics, the state of the molecule will start to change according to the time evolution equation (time-dependent Schrodinger equation)... 

As first noted by Dirac the canonical equations of motion for the real variables Xn,Pn with respect to Hmf completely equivalent to Schrodinger s equation (22) for the complex variables Moreover, it is clear that the time evolution of the nuclear DoF [Eq. (26)] can also be written as Hamilton s equations with respect to Hmf- Similarly to the equations of motion for the mapping formalism [Eqs. (60) and (61)], the mean-field equations of motion for both electronic and nuclear DoF can thus be written in canonical form. 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

In the diflfiision QMC (DMC) method [114. 119], the evolution of a trial wavefiinction (typically wavefiinctions of the Slater-Jastrow type, for example, obtained by VMC) proceeds in imaginary time, i = it, according to the time-dependent Schrodinger equation, which then becomes a drfifiision equation. All... 

Quantum Mechanical Generalities.—It will be recalled that in nonrelativistic quantum mechanics the state of a particle at a given instant t is represented by a vector in Hilbert space (f)>. The evolution of the system in time is governed by the Schrodinger equation... 

Finally, we show how to relate the modified Schrodinger equation evolution X(m) to the usual evolution T (t) [14]. Consider the modified Schrodinger equation, Eq. (12). We approximate f H) in this equation with a first-order Taylor series expansion. 

The fifth postulate stipulates that the time evolution of the state function q is determined by the time-dependent Schrodinger equation... 

A set of coupled equations for the evolution of the basis function coefficients is obtained by substituting the wavefunction ansatz of Eqs. (2.5)-(2.7) into the nuclear Schrodinger equation... 

The results of this test of the TDB-FMS method are encouraging, and we expect the gain in efficiency to be more significant for larger molecules and/or longer time evolutions. Furthermore, as noted briefly before, the approximate evaluation of matrix elements of the Hamiltonian may be improved if we can further exploit the temporal nonlocality of the Schrodinger equation. 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrodinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

If the PES are known, the time-dependent Schrodinger equation, Eq. (1), can in principle be solved directly using what are termed wavepacket dynamics [15-18]. Here, a time-independent basis set expansion is used to represent the wavepacket and the Hamiltonian. The evolution is then carried by the expansion coefficients. While providing a complete description of the system dynamics, these methods are restricted to the study of typically 3-6 degrees of freedom. Even the highly efficient multiconfiguration time-dependent Hartree (MCTDH) method [19,20], which uses a time-dependent basis set expansion, can handle no more than 30 degrees of freedom. 

As mentioned above, the correct description of the nuclei in a molecular system is a delocalized quantum wavepacket that evolves according to the Schrodinger equation. In the classical limit of the single surface (adiabatic) case, when effectively h 0, the evolution of the wavepacket density... 

Using the BO approximation, the Schrodinger equation describing the time evolution of the nuclear wave function, can be written... 

In a classical limit of the Schrodinger equation, the evolution of the nuclear wave function can be rewritten as an ensemble of pseudoparticles evolving under Newton s equations of motion... 

The time-dependent Schrodinger equation governs the evolution of a quantum mechanical system from an initial wavepacket. In the case of a semiclassical simulation, this wavepacket must be translated into a set of initial positions and momenta for the pseudoparticles. What the initial wavepacket is depends on the process being studied. This may either be a physically defined situation, such as a molecular beam experiment in which the particles are defined in particular quantum states moving relative to one another, or a theoretically defined situation suitable for a mechanistic study of the type what would happen if. .. 

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