Hamiltonian stationary

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

In the language of quanPim meehanies, the time-dependent B -field provides a perturbation with a nonvanishing matrix element joining the stationary states a) and P). If the rotating field is written in temis of an amplitude a perturbing temi in tlie Hamiltonian is obtained... 

Repeat E.verdse 9-.S using the Arguslah package and the MNDO and PM3 Hamiltonians. Deteniiine ttie total energy of atomization to ttie separated stationary atoms and electrons in hartrecs. 

This spatial distribution is not stationary but evolves in time. So in this ease, one has a wavefunetion that is not a pure eigenstate of the Hamiltonian (one says that E is a superposition state or a non-stationary state) whose average energy remains eonstant (E=E2,i ap + El 2 bp) but whose spatial distribution ehanges with time. 

The first-order eorreetion ean be thought of as arising from the response of the wavefunetion (as eontained in its ECAO-MO and Cl amplitudes and basis funetions Xv) plus the response of the Hamiltonian to the external field. Beeause the MCSCF energy funetional has been made stationary with respeet to variations in the Cj and Ci a amplitudes, the seeond and third terms above vanish ... 

In currently available software, the Hamiltonian above is nearly never used. The problem can be simplified by separating the nuclear and electron motions. This is called the Born-Oppenheimer approximation. The Hamiltonian for a molecule with stationary nuclei is... 

Since nuclei are much heavier than electrons and move slower, the Born-Oppenheimer Approximation suggests that nuclei are stationary and thus that we can solve for the motion of electrons only. This leads to the concept of an electronic Hamiltonian, describing the motion of electrons in the potential of fixed nuclei. 

In the derivation above, we have included the kinetic energy of the nuclei in the Hamiltonian and considered a stationary state. In Eq. II.3, this term has been neglected, and we have instead assumed that the nuclei have given fixed positions. It has been pointed out by Slater34 that, if the nuclei are not situated in the proper equilibrium positions, the virial theorem will appear in a slightly different form. (A variational derivation has been given by Hirschfelder and Kincaid.11)... 

If the hamiltonian is truly stationary, then the wx are the space-parts of the state function but if H is a function of t, the wx are not strictly state functions at all. Still, Eq. (7-65) defines a complete orthonormal set, each wx being time-dependent, and the quasi-eigenvalues Et will also be functions of t. It is clear on physical grounds, however, that to, will be an approximation to the true states if H varies sufficiently slowly. Hence the name, adiabatic representation. 

Interaction Representation.—In many physical problems the hamiltonian of a system that is engaged in interaction with another is of the form H + V,H being the stationary normal ( unperturbed ) hamiltonian and V the interaction. Equation (7-51) then reads... 

A special case of some interest is one in which the hamiltonian is stationary and the are taken to be its eigenfunctions. The matrix H is then diagonal Eq. (7-53) results in... 

The critical points of the equivalent classical Hamiltonian occur at stationary state energies of the quantum Hamiltonian H and correspond to stationary states in both the quantum and generalized classical pictures. These points are characterized by the constrained generalized eigenvalue equation obtained by setting the time variation to zero in Eq. (4.17)... 

It was shown in the earlier sections that the existence or nonexistence of quantum monodromy in two-dimensional maps depends on the relative dispositions of the critical points and relative equilibria of the Hamiltonian, which involves a search for the stationary points of with respect to K and Qk- For L = 0 there is a root at the critical point K = J, and other possible roots given by... 

The appearance of the Hamiltonian operator in equation (3.55) as stipulated by postulate 5 gives that operator a special status in quantum mechanics. Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator for a given system is sufficient to determine the stationary states of the system and the expectation values of any other dynamical variables. 

The stationary states of file system are described by the eigenfunctions and the eigenvalues s of the unpertuibed Hamiltonian. 

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

A first insight into a different description of a chemical process can be obtained from an analysis of a (diatomic) dissociation process. Consider the standard treatment of a stable diatomic molecule. The word stable implies already the existence of a measurable characteristic size around which the electro-nuclear system fluctuates in its ground electronic state (i.e. a stationary molecular Hamiltonian with ground state). In standard quantum chemistry, this is the nuclear equilibrium distance. 

The theoretical view advocated here focus attention on the quantum states relevant to the description of particular phenomena. The concept of stationary Hamiltonians follows from the coordinate representation and leads to a numerical determination of relevant quantum states. He will denote this class of Hamiltonians. Thus, given a system that can be decomposed into n-electrons and m-nuclei in different dispositions, the set of He =Ea(l/tna)[pa]2+Vcoui are written in terms of fluctuation coordinates around particular... 

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