Schrodinger equation in three dimensions

A particle of mass m moving in a field-free space provides the simplest application of the Schrodinger equation in three dimensions. Since V is constant (we choose the value zero for convenience), the amplitude equation 12-8 assumes the following form ... 

The quantum-mechanical part of the problem is reduced to propagating the solution to a time-dependent Schrodinger equation in three dimensions ... 

Ultimately, our goal is to solve the Schrodinger equation in three-dimensions for the electron in the hydrogen atom. This electron is subject to a potential energy term that involves a Coulombic attraction toward the nucleus. However, the solution to this differential equation is not a trivial one. We therefore choose a somewhat similar, but simpler, problem—the particle in a box—to demonstrate the procedure and to illustrate some of the principles of quantum mechanics. We then extrapolate those results to the hydrogen atom in a later chapter. 

Inside the box, where the potential energy is zero, the time-independent Schrodinger equation in three-dimension is given by Equation (3.45). Separation of the variables yields Equation (3.61) ... 

We now turn our attention to the nonperturbative regime. Due to advances in laser technology over the past decade, many experiments are now possible where the laser field is stronger than the nuclear attraction. The time-dependent field cannot be treated perturbatively, and even solving the time-dependent Schrodinger equation in three dimensions for the evolution of two interacting electrons is barely feasible with present-day computer technology. ... 

To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles (usually termed 0 and ([)). For any non-linear molecule, three angles (usually a, P, and y) are needed. Hence the rotational Schrodinger equation for a nonlinear molecule is a differential equation in three-dimensions. 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

To find the wavefunctions and energy levels of an electron in a hydrogen atom, we must solve the appropriate Schrodinger equation. To set up this equation, which resembles the equation in Eq. 9 but allows for motion in three dimensions, we use the expression for the potential energy of an electron of charge — e at a... 

In three dimensions the rotating diatomic molecule is equivalent to a particle moving on the surface of a sphere. Since V — 0 the Schrodinger equation is... 

Before starting the discussion on confined atoms, we shall briefly describe the simplest standard confined quantum mechanical system in three dimensions (3-D), namely the particle-in-a-(spherical)-box (PIAB) model [1], The analysis of this system is useful in order to understand the main characteristics of a confined system. Let us note that all other spherically confined systems with impenetrable walls located at a certain radius, Rc, transform into the PIAB model in the limit of Rc —> 0. For the sake of simplicity, we present the model in one-dimension (1-D). In atomic units (a.u.) (me=l, qc 1, and h = 1), the Schrodinger equation for an electron confined in one-dimensional box is... 

In this chapter we return to the question of the geometrical interpretation of the algebraic approach. Specifically, we need to make contact with the concept of the potential function which is central to the geometrical point of view. For example, in three dimensions, one has the Schrodinger equation (1.2)... 

The determination of Ey requires that Schrodinger s equation for the collection of atoms must be solved with all the associated nuclei and electrons in three dimensions. For a molecule with N nuclei and n electrons, the time-independent Schrodinger s equation is given by... 

Quantum mechanics describes molecules in terms of interactions among nuclei and electrons, and molecular geometry in terms of minimum energy arrangements of nuclei. All quantum mechanical methods ultimately trace back to the Schrodinger equation, which for the special case of hydrogen atom (a single particle in three dimensions) may be solved exactly. ... 

G is the reduced Green function of the Schrodinger equation and B = (Us)-Action of the operator O2 on the wave function can be checked not to produce functions more singular than G2 or c2. Therefore, in contrast to the second iteration of the original perturbation, Eq.(12), that of the operator 02 delivers a result which is finite in three dimensions. 

At this point, the two Schrodinger equations for both the singlet ground state and the first triplet excited state have to be solved in three dimensions. For that purpose, potential energy functions have to be calculated and are in progress in our laboratory. 

Up to now we have considered the properties of resonance states in one-dimensional potentials. In the subsequent sections, the discussion will focus on the decay of polyatomic molecules, potential energy surfaces of which depend at least on three variables. Let us now survey — in a more qualitative maimer — the new aspects introduced by the additional degrees of freedom. Practical issues of the solution of the Schrodinger equation in more than one dimension are reserved for Sect. 4. 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

Such a theory is embodied in the wave equation of Schrodinger. The propagation of a wave in three dimensions is represented by the expression g ... 

The general case of the rotator free to move in three dimensions is more complicated, but is treated according to similar principles. The Schrodinger equation is first expressed in spherical polar coordinates, r, 6, and . For the rotation of a rigid body about its centre of gravity, r is constant and is included in a term representing the moment of inertia, /. The conditions for physically admissible solutions lead to the result... 

Many of the models used for studying molecules with large numbers of atoms are said to be classical because they rely on assumptions that allow the use of Newton s equations for the motion of the nuclei. Coming to this subject for the first time, one may wonder why we don t use quantum mechanics, since that is a more modern and comprehensive theory of the behavior of matter. A system of N atoms can be described in a probabilistic way using the Schrodinger equation. For illustration, consider a water molecule which has 10 electrons and 3 nuclei, each of which is identified with a point in three dimensions, thus the basic object describing the probability distribution for an isolated water molecule is a complexvalued function of 39 variables. Denote a point in this space by the coordinates... 

This is the Schrodinger wave equation in one dimension. We will leave consideration of the three-dimensional Schrodinger equation until Chapter 5. 

Such quantum capture theories in three dimensions have been developed to solve the Schrodinger equation for the long-range attractive entrance channel potential in a coupled rotational-states formalism. A rotationally adiabatic approximation to this theory has been developed by constructing potential curves that describe the evolution of... 

E. Schrodinger (1926), following the earher work of L. deBroglie (1924), advanced a fundamental equation of wave mechanics. The Schrodinger equation of wave mechanics was developed for waves oscillating in three dimensions with co-ordinates x, y and z ... 

By analogy with the one-dimensional Schrodinger equation given in Equation 1.24, the Schrodinger eqnation for the wavefunction y, z) of a single electron interacting in three dimensions with a nucleus of charge +Ze is °... 

Erwin Schrodinger evaluated the theory of de Broglie and became convinced that matter waves, as many other waves, are functions in three dimensions and solutions of a differential equation. Due to the boundary conditions, there are eigenvalues. It would be reasonable if there was a correspondence between these eigenvalues and the Bohr quantum numbers. 

Again, the potential energy V for 3-D rotational motion can be set to zero, so acceptable wavefunctions for rotation in three dimensions must satisfy the Schrodinger equation, which is... 

The time-independent Schrodinger equation for a free particle moving in three dimensions is... 

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