Time-Independent Hamiltonian Operator

Normally the TDSE cannot be solved analytically and must be obtained numerically. In the numerical approach we need a method to render the wave function. In time-dependent quantum molecular reaction dynamics, the wave function is often represented using a discrete variable representation (DVR) [88-91] or Fourier Grid Hamiltonian (FGH) [92,93] method. A Fast Fourier Transform (FFT) can be used to evaluate the action of the kinetic energy operator on the wave function. Assuming the Hamiltonian is time independent, the solution of the TDSE may be written... 

The time-dependent Schrodinger equation Hif/ = ih- represents the time evolution of an arbitrary initial wave function. The assumption that translation in time is a unitary operator leads to preserving the normalization of the wave function and of the mean value of the Hamiltonian. If the Hamiltonian is time-independent, then one obtains the formal solution to the Schrodinger equation by applying the operator exp(- H) to the initial... 

The Hamiltonian remains time-independent but in the Heisenberg picture the constituent operators a aj are replaced by a,(t), aj(r). 

If the Hamiltonian operator does not contain the time variable explicitly, one can solve the time-independent Schrodinger equation... 

The first of these equations is ealled the time-independent Sehrodinger equation it is a so-ealled eigenvalue equation in whieh one is asked to find funetions that yield a eonstant multiple of themselves when aeted on by the Hamiltonian operator. Sueh funetions are ealled eigenflinetions of H and the eorresponding eonstants are ealled eigenvalues of H. 

The path-integral quantum mechanics relies on the basic relation for the evolution operator of the particle with the time-independent Hamiltonian H x, p) = -i- V(x) [Feynman and... 

To solve the time-independent Schrodinger equation for the nuclei plus electrons, we need an expression for the Hamiltonian operator. It is... 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

Since div ( ) mid div 3 (x) commute with 8(x ) and 3 t (x ) for x0 —x, they have vanishing commutators with the hamiltonian and hence, they are time-independent operators. In fact, their constancy in tame implies that they commute with 3 (x) and S(x) at all times and hence they must be c-number multiples of the unit operator. If these c-numbers are set equal to zero initially, they will remain zero for all times. With this initial choice for div 8(x) and div 3tf(x), the operators S and satisfy all of the Maxwell equations (these now are operator equations ) ... 

To facilitate the derivation we shall assume that we are in the Heisenberg picture and dealing with a time-independent hamiltonian, i.e., H(t) — 27(0) = 27, in which case Heisenberg operators at different times are related by the equation... 

Thus, the spatial function (q) is actually a set of eigenfunctions t/ n(q) of the Hamiltonian operator H with eigenvalues E . The time-independent Schrodin-ger equation takes the form... 

The variation method gives an approximation to the ground-state energy Eq (the lowest eigenvalue of the Hamiltonian operator H) for a system whose time-independent Schrodinger equation is... 

In fin problem of interest here, the Hamiltonian in Bq. (62) can be decomposed into a time-independent, unperturbed part and a much smaller, time-dependent operator H (t). Then, die Hamiltonian becomes to first oiler... 

For conservative systems with time-independent Hamiltonian the density operator may be defined as a function of one or more quantum-mechanical operators A, i.e. g= tp( A). This definition implies that for statistical equilibrium of an ensemble of conservative systems, the density operator depends only on constants of the motion. The most important case is g= [Pg.463]

The system is prepared at t=0 in the quantum state Pik> and the question is how to calculate the probability that at a later time t the system is in the state Fjn>. By construction, these quantum states are solutions of molecular Hamiltonian in absence of the radiation field, Hc->Ho Ho ik> = e k Fik> and H0 Pjn> = Sjn xPJn>. The states are orthogonal. The perturbation driving the jumps between these two states is taken to be H2(p,A)= D exp(icot), where co is the frequency of the incoherent radiation field and D will be a time independent operator. From standard quantum mechanics, the time dependent quantum state is given by ... 

These are the solutions fj of the time-independent Schrddinger equation Ht , = E/Y , where H is the Hamiltonian operator of the system and , is the energy corresponding to the state y,-. 

One simple form of the Schrodinger equation—more precisely, the time-independent, nonrelativistic Schrodinger equation—you may be familiar with is Hx i = ty. This equation is in a nice form for putting on a T-shirt or a coffee mug, but to understand it better we need to define the quantities that appear in it. In this equation, H is the Hamiltonian operator and v i is a set of solutions, or eigenstates, of the Hamiltonian. Each of these solutions,... 

The starting point for Lowdin s PT [1-6] and Eeshbach s projection formalism [7-9] is the fragmentation of the Hilbert space H = Q V, of a given time-independent Hamiltonian H, into subspaces Q and V by the action of projection operators Q and P, respectively. The projection operators satisfy the following conditions ... 

The initial and final asymptotic states are always expanded in the time independent basis associated with the molecular hamiltonian the scattering matrix is unitary. Note again that the basis contains all possible resonance and compound states. If there is no interaction, the scattering matrix is the unit matrix 1. Formally, one can write this matrix as S= 1+iT where T is an operator describing the non-zero scattering events including chemical reactions. Thus, for a system prepared in the initial state Op, the probability amplitude to get the system in the... 

One point which has not been addressed in the example of the time-independent harmonic oscillator is the non-perturbative treatment of the time dependence in the system Hamiltonians. Both the TL and the TNL non-Markovian theories employ auxiliary operators or density matrices, respectively, and can be applied in strongly driven systems [29,32]. This point will be shown to be very important in the examples for the molecular wires under the influence of strong laser fields. 

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