Schrodingers Time-Independent Wave Equation

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

D. J. Kouri, Y. Huang, W. Zhu, and D. K. Hoffman, Variational principles for the time-independent wave-packet-Schrodinger and wave-packet-Lippmann-Schwinger equations, J. Chem. Phys. 100 3662 (1994). 

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. 

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. 

Time-independent wave packet Schrodinger equation... 

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. 

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]... 

The relationship between the time-dependent and time-independent approaches, given by a Fourier transform between G(e) and U t), was elucidated by Bloch [43]. Of course, in either approach, the final result is time-independent, since, after all, we are solving a time-independent or stationary Schrodinger equation. It would thus seem that the time-independent approach—as followed by Hugenholtz—would turn out to be more natural and simpler. This is indeed the case when we are primarily interested in the energy. However, in order to elucidate the PT structure of the exact wave function, the time-dependent approach is beneficial cf. Ref. [34]). 

The basis of the use of quantum mechanics is that we can construct a total molecular wave function using which a time independent non-relativistic Schrodinger equation of the form. 

The problem is now to find which time-independent wave functions describe the energy states of the electron in a H atom. In quantum mechanics, these wave functions are found as solutions to the Schrodinger equation ... 

The stationary-state wave functions and energy levels of a one-particle, one-dimensional system are found by solving the time-independent Schrbdinger equation (1.19). In this chapter, we solve the time-independent Schrodinger equation for a very simple system, a particle in a one-dimensional box (Section 2.2). Because the Schrbdinger equation is a differential equation, we first discuss differential equations. 

If the potential V is time-independent, wave functions satisfying the Schrodinger equation can be factored into a space function and one periodic in time. 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

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. 

Various difficulties of classical physics, including inadequate description of atoms and molecules, led to new ways of visualizing physical realities, ways which are embodied in the methods of quantum mechanics. Quantum mechanics is based on the description of particle motion by a wave function, satisfying the Schrodinger equation, which in its time-independent form is ... 

Just like any spectroscopic event EPR is a quantum-mechanical phenomenon, therefore its description requires formalisms from quantum mechanics. The energy levels of a static molecular system (e.g., a metalloprotein in a static magnetic field) are described by the time-independent Schrodinger wave equation,... 

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 ... 

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... 

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