Hamiltonian operator, Schrodinger

When the Hamilton function had been replaced by the Hamiltonian operator, SchrOdinger was able to write down a time-independent (Schrodinger) equation (SE) for standing waves as a function of the electron coordinates ... 

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

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

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

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

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... 

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... 

Perturbation theory provides a procedure for finding approximate solutions to the Schrodinger equation for a system which differs only slightly from a system for which the solutions are known. The Hamiltonian operator H for the system of interest is given by... 

In reality, this term is not small in comparison with the other terms so we should not expect the perturbation technique to give accurate results. With this choice for the perturbation, the Schrodinger equation for the unperturbed Hamiltonian operator may be solved exactly. 

A theory for nonequilibrium quantum statistical mechanics can be developed using a time-dependent, Hermitian, Hamiltonian operator Hit). In the quantum case it is the wave functions [/ that are the microstates analogous to a point in phase space. The complex conjugate / plays the role of the conjugate point in phase space, since, according to Schrodinger, it has equal and opposite time derivative to v /. 

The nonrelativistic, electronic Schrodinger Hamiltonian operator, designated as H, is represented by... 

To solve for a Schrodinger s operator method will be used by way of illustration. The Hamiltonian operator is factored into two operators containing first derivatives only, using the relationship2... 

In the Schrodinger picture operators in the case of a closed system do not depend explicitly on the time, but the state vector is time dependent. However, the expectation values are generally functions of the time. The commutator of the Hamiltonian operator H= —(h/2iri)(d/dt) and another operator A, is defined by... 

The solution of the unperturbed Hamiltonian operator forms a complete orthonormal set. The perturbed Schrodinger equation is given by... 

The Schrodinger equation for nuclear motion contains a Hamiltonian operator Hop,nuc consisting of the nuclear kinetic energy and a potential energy term which is Eeiec(S) of Equation 2.7. Thus... 

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,... 

Our main concern in this section is with the actual propagation forward in time of the wavepacket. The standard ways of solving the time-dependent Schrodinger equation are the Chebyshev expansion method proposed and popularised by Kossloff [16,18,20,37 0] and the split-operator method of Feit and Fleck [19,163,164]. I will not discuss these methods here as they have been amply reviewed in the references just quoted. Comparative studies [17-19] show conclusively that the Chebyshev expansion method is the most accurate and stable but the split-operator method allows for explicit time dependence in the Hamiltonian operator and is often faster when ultimate accuracy is not required. All methods for solving the time propagation of the wavepacket require the repeated operation of the Hamiltonian operator on the wavepacket. It is this aspect of the propagation that I will discuss in this section. 

The first of these equations is called the time-independent Schrodinger equation it is a so-called eigenvalue equation in which one is asked to find functions that yield a constant multiple of themselves when acted on by the Hamiltonian operator. Such functions are called eigenfunctions of H and the corresponding constants are called eigenvalues of H. 

We ve derived a complete many-electron Hamiltonian operator. Of course, the Schrodinger equation involving it is intractable, so let s consider a simpler problem, involving the one-electron hamiltonian... 

T. Kato, On the fundamental properties of Hamiltonian operators of Schrodinger type, Trans. Amer. Math. Soc. (1951), in press. 

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