Time-independent Hamiltonian

A time-varying wave function is also obtained with a time-independent Hamiltonian by placing the system initially into a superposition of energy eigenstates ( n)), or forming a wavepacket. Frequently, a coordinate representation is used for the wave function, which then may be written as... 

In this method, one notes that real-valued solutions of the time-independent Hamiltonian of a 2 x 2 matrix form can be written in terms of an 0(<1), q), which is twice the mixing angle, such that the electronic component which is initially 1 is cos [0(4>, < )/2], while that which is initially 0 is sin [0(4>,<3r)/2]. For the second matrix form in Eq. (68) (in which, for simplicity f x) = 1), we get... 

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... 

The parameters e called quasienergies play the same role for periodic motion as the usual energies do for time-independent Hamiltonians. By using the definition (5.6) it is easy to obtain [Benderskii and Makarov 1992]... 

The Schrodinger equation with its time-independent hamiltonian does not in fact constitute a dynamical theorem it is simply a description of the time-dependence of the probability field corresponding to steady states or equilibrium conditions. 

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... 

In the derivation of response functions one considers a molecule or an atom described by the time-independent Hamiltonian which is perturbed by an external one-electron perturbation V t e). 

The signal detectable after a pulse of duration Tpulse, is given by (25), which for the time-independent Hamiltonian (i.e., under the static condition) yields... 

The Hamiltonian will in general be a function of time, but the special case of a time-independent Hamiltonian is of central importance. Suppose that H has an eigenvalue of E and an eigenvector v at t = 0. Suppose that the state of the system at any other time t is u(t) = wexp(—iat), where the argument of the phase factor is to be established. The derivative... 

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]

As for classical systems, measurement of the properties of macroscopic quantum systems is subject to experimental error that exceeds the quantum-mechanical uncertainty. For two measurable quantities F and G the inequality is defined as AFAG >> (5F6G.The state vector of a completely closed system described by a time-independent Hamiltonian H, with eigenvalues En and eigenfunctions is represented by... 

Consider a spin system whose spin Hamiltonian consists of a time-independent Hamiltonian H0 and a stochastic perturbation Hamiltonian H,(t) due to a small spin-lattice coupling,... 

When a molecular system is placed in static and/or dynamic external electric fields, a perturbation term has to be added to the unperturbed time independent Hamiltonian, Hg ... 

The total Hamiltonian reads 7i(0 = Hq + V t), where Hq is the time-independent Hamiltonian of the unperturbed system and V(t) is the interaction describing the coupling between field and electron. In the dipole approximation. 

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 ... 

When F in (2.64) is a time-independent Hamiltonian, then (2.64) is the time-independent Schrodinger equation H n = En n and the energies satisfy (2.68) ... 

In spectroscopy, we have a system (atom or molecule) that starts in some stationary state of definite energy, is exposed to electromagnetic radiation for a limited time, and is then found to be in some other stationary state. Let H0 be the time-independent Hamiltonian of the system in the absence of radiation. We have... 

Because of interelectronic repulsions, the Schrodinger equation for many-electron atoms and molecules cannot be solved exactly. The two main approximation methods used are the variation method and perturbation theory. The variation method is based on the following theorem. Given a system with time-independent Hamiltonian //, then if

well-behaved function that satisfies the boundary conditions of the problem, one can show (by expanding

[Pg.271]

Show that addition of a constant C to a time-independent Hamiltonian H leaves the stationary-state wave functions unchanged and adds C to each energy eigenvalue. 

We note that the separation into the three types of transitions (7), (2), and (2) is somewhat artificial. In fact, molecular collisions and transitions due to external fields are special examples of prepared states. Time evolution of a system described by a time-independent Hamiltonian does occur in general, unless the initial state of the system is described by a ket which is an eigenket to the complete Hamiltonian. 

In the limit where the nuclear Zeeman term in the nuclear spin hamiltonian is much larger than the quadrupole interaction, it is only the secular part of Hq that contributes to the time-independent hamiltonian, H0. 

For a time-independent Hamiltonian, the wavefunction satisfying the time-dependent Schrodinger equation contains a multiplicative factor exp(-iEt/h). For positronium this time-dependent factor leads to a probability density proportional to... 

The two first terms form the time-independent Hamiltonian of the quantum subsystem where H0 is the Hamiltonian of molecular subsystem in vacuum and the operator WqM/cm represents the coupling between the molecular system and the structured environment. 

Given the basic constituents of a material system, namely, the number of electron and nuclei and a time-independent Hamiltonian H with a complete set of basis states j and eigenvalues c,, an arbitrary quantum state is given as a linear superposition with complex number amplitudes ... 

Let us consider a system in equilibrium, described in the absence of external perturbations by a time-independent Hamiltonian Ho. We will be concerned with equilibrium average values which we will denote as (...), where the symbol (...) stands for Trp0... with p0 = e H"/ Vre the canonical density operator. Since we intend to discuss linear response functions and symmetrized equilibrium correlation functions genetically denoted as Xba(, 0 and CBA t,t ), we shall assume that the observables of interest A and B do not commute with Ho (were it the case, the response function %BA(t, t ) would indeed be zero). This hypothesis implies in particular that A and B are centered A) =0,... 

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