Hamiltonian Schrodinger equation

Eigen function In wave mechanics, the Schrodinger equation may be written using the Hamiltonian operator H as... 

This establishes our assertion that the former roots are overwhelmingly more numerous than those of the latter kind. Before embarking on a formal proof, let us illustrate the theorem with respect to a representative, though specific example. We consider the time development of a doublet subject to a Schrodinger equation whose Hamiltonian in a doublet representation is [13,29]... 

If the Hamiltonian operator contains the time variable explicitly, one must solve the time-dependent Schrodinger equation... 

The technique for this calculation involves two steps. The first step computes the Hamiltonian or energy matrix. The elements of this matrix are integrals involving the atomic orbitals and terms obtained from the Schrodinger equation. The most important con-... 

HyperChem s semi-empirical calculations solve (approximately) the Schrodinger equation for this electronic Hamiltonian leading to an electronic wave function I eiecW for the electrons ... 

Solutions to a Schrodinger equation for this last Hamiltonian (7) describe the vibrational, rotational, and translational states of a molecular system. This release of HyperChem does not specifically explore solutions to the nuclear Schrodinger equation, although future releases may. Instead, as is often the case, a classical approximation is made replacing the Hamiltonian by the classical energy ... 

Molecular quantum mechanics finds the solution to a Schrodinger equation for an electronic Hamiltonian, Hgjg., that gives a total energy, Egjg(-(R) + V (R,R). Repeated solutions at different nuclear configurations, R, lead to some approximate potential energy sur-... 

Rather than solve a Schrodinger equation with the Nuclear Hamiltonian (above), a common approximation is to assume that atoms are heavy enough so that classical mechanics is a good enough approximation. Motion of the particles on the potential surface, according to the laws of classical mechanics, is then the subject of classical trajectory analysis or molecular dynamics. These come about by replacing Equation (7) on page 164 with its classical equivalent ... 

This Hamiltonian is then used in the Schrodinger equation describing the motion of electrons in the field of fixed nuclei ... 

This Hamiltonian is used in the Schrodinger equation for nuclear motion, describing the vibrational, rotational, and translational states of the nuclei. Solving the nuclear Schrodinger equation (at least approximately) is necessary for predicting the vibrational spectra of molecules. 

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

So, let s get a bit more chemical and imagine the formation of an H2 molecule from two separated hydrogen atoms, Ha and Hb, initially an infinite distance apart. Electron 1 is associated with nucleus A, electron 2 with nucleus B, and the terms in the electronic Hamiltonian / ab, ba2 and are all negligible when the nuclei are at infinite separation. Thus the electronic Schrodinger equation becomes... 

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. 

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... 

The term d2iji/dx 2 can be thought of as a measure of how sharply the wavefunction is curved. The left-hand side of the Schrodinger equation is commonly written Hv i, where H is called the hamiltonian for the system then the equation takes the deceptively simple form... 

E. Quantitative Aspects of Tq-S Mixing 1. The spin Hamiltonian and Tq-S mixing A basic problem in quantum mechanics is to relate the probability of an ensemble of particles being in one particular state at a particular time to the probability of their being in another state at some time later. The ensemble in this case is the population distribution of nuclear spin states. The time-dependent Schrodinger equation (14) allows such a calculation to be carried out. In equation (14) i/ (S,i) denotes the total... 

Perhaps the most straightforward method of solving the time-dependent Schrodinger equation and of propagating the wave function forward in time is to expand the wave function in the set of eigenfunctions of the unperturbed Hamiltonian [41], Hq, which is the Hamiltonian in the absence of the interaction with the laser field. 

The one-electron Hamiltonian operator h = h +v, with kinetic energy operator, generates a complete spectrum of orbitals according to the Schrodinger equation... 

The perturbation theory is the convenient starting point for the determination of the polarizability from the Schrodinger equation, restricted to its electronic part and the electric dipole interaction regime. The Stark Hamiltonian —p. describes the dipolar interaction between the electric field and the molecule represented by its... 

If the potential energy of a system is an even function of the coordinates and if (q) is a solution of the time-independent Schrodinger equation, then the function is also a solution. When the eigenvalues of the Hamiltonian... 

In order to apply quantum-mechanical theory to the hydrogen atom, we first need to find the appropriate Hamiltonian operator and Schrodinger equation. As preparation for establishing the Hamiltonian operator, we consider a classical system of two interacting point particles with masses mi and m2 and instantaneous positions ri and V2 as shown in Figure 6.1. In terms of their cartesian components, these position vectors are... 

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