THREE-DIMENSIONAL PROBLEMS

THREE-DIMENSIONAL PROBLEMS

We have discussed the Schrodinger equation and its applications only for one dimensional problems, the analog of a Newtonian particle moving along a straight line. The straight-line model is adequate for some applications, but to understand atomic structure, we need a three-dimensional generalization. It’s not difficult-to guess what the three-dimensional Schrodinger equation should look like. First, the wave function ljI is a function of all three space coordinates (x, y, z). In general, the potential-energy function also depends on all three coordinates and can be written as Utx, y, z). Next, recall that we obtained the term containing d2lj1/d:i in the one-dimensional equation, Eq. (42-17), by a line of reasoning based on the relation of kinetic energy K to momentum p: K =/l2m. If the particle’s momentum has three components (Px’ Py’ p,), then the corresponding relation in three dimensions is

THREE-DIMENSIONAL PROBLEMS
THREE-DIMENSIONAL PROBLEMS

In this equation it is understood that ljI is, and U may be, a function of x, y, and z (hence the partial-derivative notation), while E is constant for a state of particular energy. We won’t pretend that we have derived Eq. (42-32). Like the one-dimensional version, this equation has to be tested by comparison of its predictions with experimental results. As we will see in later chapters, Eq. (42-32) passes this test with flying colors, so we are confident that it is the correct equation. In many practical problems, in atomic structure and elsewhere, the potential-energy function is spherically symmetric; it depends only on the distance r= (i +l +i)lfl from the origin of coordinates. To take advantage of this symmetry, we use spherical coordinates (Fig. 42-22) instead of Cartesian coordinates (x, y, z). Then a spherically symmetric potential-energy function is a function only of r, not of 9 or ;, so U = U(r). This fact turns out to simplify greatly the problem of finding solutions of the Schrodinger equation, even though the derivatives in Eq. (42-32) are considerably more complex when expressed in terms of spherical coordinates. Be careful; many math texts exchange the angles 9 and ; shown in Fig. 42-22. For the hydrogen atom the potential-energy function U(r) is the familiar Coulomb’slaw function:

We will find that for all spherically symmetric potential-energy functions U(r), each possible wave function can be expressed as a product of three functions: one a function only of r, one only of 9, and one only of ;. Furthermore, the functions of 9 and ; are the same for every spherically symmetric potential-energy function. This result is directly related to the problem of finding the possible values of angular momentum for the various states. We’]] discuss these matters more in the next chapter.

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