Time-independent wave equation

The wave function is governed by a differential equation which can also be written in the same form for light, namely the (time-independent) wave equation of Sghrodinger (1926) ... 

Equation (2) is thus the characteristic time-independent wave equation which we shall now apply to an electron by means of De Broglie s relation ... 

The stationary solutions are eigenfunctions of the time-independent wave equation (7), characterized by constant Vq. For an atom in an s-state (or any V o-state) the wave function is real, which means that the electron is at rest. This result may seem surprising, because classically a dynamic equilibrium is advanced to explain why the potential does not cause the particle to fall... 

The answer to the question of what stable electron waves are allowed in any chemical structure is given by the time-independent Schrodinger wave equation for that stmcture. (The time-independent wave equation is used to obtain the stable electron waves around atoms and other chemical structures, and the time-depen-dent wave equation is used for calculations of electron waves as they undergo transitions from one wave into another). The Schrodinger equation is not derivable directly from any previous equations it combines ideas of wave and particle behaviour that were previously considered mutually exclusive. This combination of particle and wave properties can be illustrated by discussion of the equation for the hydrogen atom. 

This is the classical time-independent wave equation for a string. 

For three-dimensional systems, the classical time-independent wave equation for an isotropic and uniform medium is... 

Figure 1-5 Solutions for the time-independent wave equation in one dimension with boundary conditions = j/(L)= 0.

Section 1-9 Schr6dinger s Time-Independent Wave Equation... 

Earlier we saw that we needed a wave equation in order to solve for the standing waves pertaining to a particular classical system and its set of boundary conditions. The same need exists for a wave equation to solve for matter waves. Schrodinger obtained such an equation by taking the classical time-independent wave equation and substituting de Broglie s relation for A. Thus, if... 

Equation (1-49) is Schrodinger s time-independent wave equation for a single particle of mass m moving in the three-dimensional potential field V. 

The boundary conditions are in general of the mixed type involving a combination of the function value and derivative at the two boundaries taken here to occur tx = a andx = b. Special cases of this equation lead to many classical functions such as Bessel functions, Legendre polynomials, Hemite polynomials, Laguerre polynomials and Chebyshev polynomials. In addition the Schrodinger time independent wave equation is a form of the Sturm-Liouville problem. 

Kouri D J, Huang Y, Zhu W and Hoffman D K 1994 Variational principles for the time-independent wave-packet-Schrddinger and wave-packet-Lippmann-Schwinger equations J. Chem. Phys. 100... 

The time-independent Schrbdinger equation describes the particle-wave duality, the square of the wave function giving the probability of finding the particle at a given position. 

Solve the time-independent Schrodinger equation for this particle to obtain the energy levels and the normalized wave functions. (Note that the boundary conditions are different from those in Section 2.5.)... 

The first two chapters serve as an introduction to quantum theory. It is assumed that the student has already been exposed to elementary quantum mechanics and to the historical events that led to its development in an undergraduate physical chemistry course or in a course on atomic physics. Accordingly, the historical development of quantum theory is not covered. To serve as a rationale for the postulates of quantum theory, Chapter 1 discusses wave motion and wave packets and then relates particle motion to wave motion. In Chapter 2 the time-dependent and time-independent Schrodinger equations are introduced along with a discussion of wave functions for particles in a potential field. Some instructors may wish to omit the first or both of these chapters or to present abbreviated versions. 

Technically, the time-independent Schrodinger equation (2) is solved for clamped nuclei. The Hamiltonian is broken into its electronic part, He, including the nuclear Coulomb repulsion energy, and the nuclear Hamiltonian HN. At this level, mass polarization effects are usually neglected. The wave function is therefore factorized as usual (r,X)= vP(r X)g(X). Formally, the electronic wave function d lnX) and total electronic energy, E(X), are obtained after solving the equation for each value of X ... 

Further Analysis of Solutions to the Time-Independent Wave Packet Equations of Quantum Dynamics II. Scattering as a Continuous Function of Energy Using Finite, Discrete Approximate Hamiltonians. 

Variational Principles for the Time-Independent Wave-Packet-Schrodinger and Wave-Packet-Lippmann-Schwinger Equations. 

This is called the radial wave equation. Apart from the term involving l, it is the same as the one-dimensional time-independent Schrodinger equation, a fact that will be useful in its solution. The last term is referred to as the centrifugal potential, that is, a potential whose first derivative with respect to r gives the centrifugal force. 

In quantum mechanics the stationary states of a system are described by the state function (or wave function) ip( x ), which satisfies the time-independent Schrodinger equation... 

The modification of the radial functions is obvious because the atomic potential V(r) will modify the spherical Bessel functions j/jcr) which belong to a free plane wave. Also, the dependence on products of k and r is lost. The RK/r) functions follow as regular solutions from the time-independent Schrodinger equation ... 

Wave packet A localized wave function, consisting of a non-stationary superposition of eigenfunctions of the time-independent Schrodinger equation. 

Thus, in BO MD the fully converged electronic wave-function and the forces acting on the nuclei must be dermined at each timestep by solving the time-independent Schrodinger equation. The time consuming evaluation of the wave-function and the gradients is the main drawback of the BO MD approach (see Figure 4-1). 

Presently, we assume that we have a time-dependent wave function, 10(/)>, and that it is normalized to unity. Furthermore, we require that 10(/)> reduces to the time-independent wave function, O), in the limit of no perturbation. The time-independent wave function, O), is the solution to the time-independent Schrodinger equation and 0) is normalized. Therefore, for an exact state we write the time-dependent wave function as [50,51]... 

Note that we do not need to solve Eq. (71) to obtain the wave function T ) from Eq. (67). This is because the whole sequence Q ,fc can also be generated recursively using the already obtained sets an, j3n and uk. To this end, we substitute Eq. (67) into the time-independent Schrodinger equation (38) and... 

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