Cartesian coordinates Schrodinger equation

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

Multiplication of q. (51) by yp x, y, z) yields the SchrOdinger equation.for the relative movement of the two particles. However, the Cartesian coordinates employed are not natural for this problem. In particular, if, as has been... 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

Many problems in nuclear physics and chemistry involve potentials, such as the Coulomb potential, that are spherically symmetric. In these cases, it is advantageous to express the time-independent Schrodinger equation in spherical coordinates (Fig. E.6). The familiar transformations from a Cartesian coordinate system (x, y, z) to spherical coordinates (r, 0, tp) are (Fig. E.6)... 

The standard approach to solving the Schrodinger equation for hydrogenlike atoms involves transforming it from Cartesian (x, y, z) to polar coordinates (r, 6, (p), since these accord more naturally with the spherical symmetry of the system. This makes it possible to separate the equation into three simpler equations, fir) = 0, fid) = 0, and fiip) = 0. Solution of the fir) equation gives rise to the n quantum number, solution of the/(0) equation to the l quantum number, and solution of the fifi) equation to the mm (often simply called m) quantum number. For each specific n = n, 1 = 1 and mm = inm there is a mathematical function obtained by combining the appropriate fir), fi fJ) and /([Pg.101]

The function ij/(r, 9, p) (clearly ij/ could also be expressed in Cartesians), depends functionally on r, 6, p and parametrically on n, l and inm for each particular set (n. I, mm ) of these numbers there is a particular function with the spatial coordinates variables r, 6, p (or x, y, z). A function like /rsiiir is a function of x and depends only parametrically on k. This ij/ function is an orbital ( quasi-orbit the term was invented by Mulliken, Section 4.3.4), and you are doubtless familiar with plots of its variation with the spatial coordinates. Plots of the variation of ij/2 with spatial coordinates indicate variation of the electron density (recall the Bom interpretation of the wavefunction) in space due to an electron with quantum numbers n, l and inm. We can think of an orbital as a region of space occupied by an electron with a particular set of quantum numbers, or as a mathematical function ij/ describing the energy and the shape of the spatial domain of an electron. For an atom or molecule with more than one electron, the assignment of electrons to orbitals is an (albeit very useful) approximation, since orbitals follow from solution of the Schrodinger equation for a hydrogen atom. 

If we place the nucleus of the hydrogen atom at the origin of a set of Cartesian coordinates, the position of the electron would be given by x, y, and z, as shown in Fig. 2.1.1. However, the solution of the Schrodinger equation for this system becomes intractable if it is done in Cartesian coordinates. Instead, this problem is solved using polar spherical coordinates r, 6, and (f>, which are also shown in Fig. 2.1.1. These two sets of coordinates are related by ... 

Because it is more convenient mathematically, the coordinate system is changed from Cartesian to spherical polar coordinates (see Fig. 12.15) before the Schrodinger equation is solved. In the system of spherical polar coordinates a given point in space, specified by values of the Cartesian coordinates x, y, and z, is described by specific values of r, 6, and < >. 

The solutions of the Schrodinger equation for the first three energy levels of the hydrogen atom are given in Table IV. The wave functions are expressed in terms of the spherical coordinates, r, the distance of the electron from the nucleus and the angles 6 and (f>. The relationship between spherical and Cartesian coordinates is seen in Figure 2. 

As we move from one-electron to many-electron atoms, both the Schrodinger equation and its solutions become increasingly complicated. The simplest many-electron atom, helium (He), has two electrons and a nuclear charge of +2e. The positions of the two electrons in a helium atom can be described using two sets of Cartesian coordinates, (xi, yi, Zi) and (xi, yz, Zz), relative to the same origin. The wave function tf depends on all six of these variables if = (x, y, Zu Xz, yz Zz)-... 

If a molecule has certain symmetry properties, important predictions about the solutions of the electronic Schrodinger equation can be made without having to solve the equation itself. Consider the case of a planar molecule, i.e. of a molecule whose nuclei lie in a plane. This plane is a symmetry plane for the molecule, and it can be shown that any eigenfunction is either symmetric or antisymmetric with respect to this plane. If one chooses the plane of the nuclei as the (y, z) plane of a Cartesian coordinate system, this means that... 

Dirac showed in 1928 that a fourth quantum number associated with intrinsic angular momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heuristically. In general, the wavefunction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Schrodinger equation and involves only the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A common shorthand notation is often used, whereby... 

Usually, of course, we are concerned with three-dimensional rather than onedimensional problems. If we decide to work with Cartesian coordinates, the potential energy will be a function of x, y, and z, and can be written as U x,y,z). The Schrodinger equation now takes the form... 

The potential energy depends only on the distance r, and not on angles that is, it is symmetrical about the nucleus. This suggests that it will be more convenient to work with polar coordinates r, 0, and 4>, which are shown in Figure 1.3. These are related to the Cartesian coordinates in the manner shown in the figure, and the Schrodinger equation in polar coordinates is... 

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