Hydrogen Schrodinger wave equation

In 1926 Erwin Schrodinger (1887-1961), an Austrian physicist, made a major contribution to quantum mechanics. He wrote down a rather complex differential equation to express the wave properties of an electron in an atom. This equation can be solved, at least in principle, to find the amplitude (height) of the electron wave at various points in space. The quantity ip (psi) is known as the wave function. Although we will not use the Schrodinger wave equation in any calculations, you should realize that much of our discussion of electronic structure is based on solutions to that equation for the electron in the hydrogen atom. 

The quantum number ms was introduced to make theory consistent with experiment. In that sense, it differs from the first three quantum numbers, which came from the solution to the Schrodinger wave equation for the hydrogen atom. This quantum number is not related to n, , or mi. It can have either of two possible values ... 

Erwin Schrodinger developed an equation to describe the electron in the hydrogen atom as having both wavelike and particle-like behaviour. Solution of the Schrodinger wave equation by application of the so-called quantum mechanics or wave mechanics shows that electronic energy levels within atoms are quantised that is, only certain specific electronic energy levels are allowed. 

Atomic orbitals are actually graphical representations for mathematical solutions to the Schrodinger wave equation. The equation provides not one, but a series of solutions termed wave functions t[ . The square of the wave function, is proportional to the electron density and thus provides us with the probability of finding an electron within a given space. Calculations have allowed us to appreciate the shape of atomic orbitals for the simplest atom, i.e. hydrogen, and we make the assumption that these shapes also apply for the heavier atoms, like carbon. 

In 1926, Erwin Schrodinger made use of the wave character of the electron and adapted a previously known equation for three-dimensional waves to the hydrogen atom problem. The result is known as the Schrodinger wave equation for the hydrogen atom, which can be written as... 

While a great deal of progress has proved possible for the case of the hydrogen atom by direct solution of the Schrodinger wave equation, some of which will be summarized below, at the time of writing the treatment of many-electron atoms necessitates a simpler approach. This is afforded by the semi-classical Thomas-Fermi theory [4-6], the first explicit form of what today is termed density functional theory [7,8]. We shall summarize below the work of Hill et al. [9], who solved the Thomas-Fermi (TF) equation for heavy positive ions in the limit of extremely strong magnetic fields. This will lead naturally into the formulation of relativistic Thomas-Fermi (TF) theory [10] and to a discussion of the role of the virial in this approximate theory [11]. 

The Schrodinger wave equation In 1926, Austrian physicist Erwin Schrbdinger (1887-1961) furthered the wave-particle theory proposed by de Broglie. Schrbdinger derived an equation that treated the hydrogen atom s electron as a wave. Remarkably, Schrbdinger s new model for the hydrogen atom seemed to apply equally well to atoms of other elements—an area in which Bohr s model failed. The atomic model in which electrons are treated as waves is called the wave mechanical model of the atom or, more commonly, the quantum mechanical model of the atom. Like Bohr s model,... 

The equation is used to describe the behaviour of an atom or molecule in terms of its wave-like (or quantum) nature. By trying to solve the equation the energy levels of the system are calculated. However, the complex nature of multielectron/nuclei systems is simplified using the Born Oppenheimer approximation. Unfortunately it is not possible to obtain an exact solution of the Schrodinger wave equation except for the simplest case, i.e. hydrogen. Theoretical chemists have therefore established approaches to find approximate solutions to the wave equation. One such approach uses the Hartree Fock self-consistent field method, although other approaches are possible. Two important classes of calculation are based on ab initio or semi-empirical methods. Ab initio literally means from the beginning . The term is used in computational chemistry to describe computations which are not based upon any experimental data, but based purely on theoretical principles. This is not to say that this approach has no scientific basis — indeed the approach uses mathematical approximations to simplify, for example, a differential equation. In contrast, semi-empirical methods utilize some experimental data to simplify the calculations. As a consequence semi-empirical methods are more rapid than ab initio. 

The Schrodinger wave equation that describes the motion of an electron in an isolated hydrogen atom is a second-order linear differential equation that may be solved after specification of suitable boundary conditions, based on physical considerations. The solution to the equation, known as a wave function provides an exhaustive description of the dynamic variables associated with electronic motion in the central Coulomb field of the proton. 

Information about the wavefunction is obtained from the Schrodinger wave equation, which can be set up and solved either exactly or approximately the Schrodinger equation can be solved exactly only for a species containing a nucleus and only one electron (e.g. H, He ), i.e. a hydrogen-like system. 

Mathematically, P describes the motion of an electron in an orbital. The modulus of the wave function squared, l P(r)P, is a direct measure of the probability of finding the electron at a particular location. The Schrodinger wave equation can be solved exactly for hydrogen. To apply it you must first transform it into polar coordinates (r,0,< )) and then solve using the method of separation of variables (described in, e.g., Kreyszig, 1999). 

The Schrodinger wave equation can be set for atoms with more than one electron, but it cannot be solved exactly in these cases. The second and subsequent electrons introduce the complicating feature of electron-electron repulsion. Nevertheless, the basic characteristics of the orbitals do not change and the results obtained for hydrogen are applied to many-electron atoms. 

Formulation of the Schrodinger Wave Equation for Hydrogen-like Atoms... 

The Schrodinger wave equation can only be used for a system with one electron, such as the hydrogen atom and cannot solve for larger atoms and molecules. Two approximations are used the valence bond and the molecular orbital methods. 

The spatial orientations of the atomic orbitals of the hydrogen atom are very important in the consideration of the interaction of orbitals of different atoms in the production of chemical bonds. The solutions of the Schrodinger wave equation for the hydrogen atom may be represented by the equation ... 

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. 

In principle, we can perform some sort of molecular orbital calculation on molecules of almost any complexity. It is, however, often extremely profitable to relate the properties of a complex system to those of a simpler one. Take, for example, the hydrogen atom in an electric field. It is much more instructive to see how the unperturbed levels of the atom are altered as a field is applied, than to solve the Schrodinger wave equation for the more complex case of the molecule with the field on. Analogously, to appreciate the orbital structure of complex systems it is much more insightful to start off with the levels of a simpler one and switch on a perturbation. 3.1-3.3 show three examples of different types of perturbations... 

The following relationships involving the three quantum numbers arise from the solution of the Schrodinger wave equation for the hydrogen atom. In this solution the values of the quantum numbers are fixed in the order listed. 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

Stark effect of a hydrogen-like atom, using the Schrodinger wave mechanics. Their equation, obtained independently and by different methods, is... 

In his first communication23 on the new wave mechanics, Schrodinger presented and solved his famous Eq. (1.1) for the one-electron hydrogen atom. To this day the H atom is the only atomic or molecular species for which exact solutions of Schrodinger s equation are known. Hence, these hydrogenic solutions strongly guide the search for accurate solutions of many-electron systems. 

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