THREE DIMENSIONAL SHROEDINGER EQUATION PHYSICS 420 SPRING 2006 Dennis Papadopoulos LECTURE 20 THREE DIMENSIONAL SHROEDINGER EQUATION

Figure 8. 1 A particle confined to move in a cubic box of sides L Figure 8.1 A particle confined to move in a cubic box of sides L. Inside the box U = 0. The potential energy is infinite at the walls and outside the box. Fig. 8-1, p.261

Figure 8.2 Change in particle velocity (or momentum) during collision with a wall of the box. For elastic collision with a smooth wall, the component normal to the wall is reversed, but the tangential components are unaffected. Fig. 8-2, p.261

Figure 8.3 An energy-level diagram for a particle confined to a cubic box. The ground-state energy is E0 = 32h2/2mL2, and n2 = n1 2 + n2 2 + n3 2. Note that most of the levels are degenerate. Fig. 8-3, p.264

Table 8-1, p.265

Figure 8.4 Probability density (unnormalized) for a particle in a box: (a) ground state, 1112; (b) and (c) first excited states, 2112 and 1212. Plots are for 2 in the plane z =1/2 L. In this plane, 1122(not shown) is indistinguishable from 1112. Fig. 8-4, p.265

28.14 MeV 4.05 MeV Figure 7.9 (a)  decay of a radioactive nucleus. (b) The potential energy seen by an  particle emitted with energy E. R is the nuclear radius, about 10-14 m, or 10 fm.  particles tunneling through the potential barrier between R and R1 escape the nucleus to be detected as radioactive decay products. 30 MeV KE Fig. 7-9, p.245

Figure 8.5 The central force on an atomic electron is one directed toward a fixed point, the nucleus. The coordinates of choice here are the spherical coordinates r, ,  , centered on the nucleus. Fig. 8-5, p.266

Figure 8.6 The angular momentum L of an orbiting particle is perpendicular to the plane of the orbit. If the direction of L were known precisely, both the coordinate and momentum in the direction perpendicular to the orbit would be known, in violation of the uncertainty principle. Fig. 8-6, p.267

Table 8-2, p.269

Table 8-3, p.269

Table 8-4, p.280

Table 8-5, p.280

Figure 8. 8 Energy-level diagram of atomic hydrogen Figure 8.8 Energy-level diagram of atomic hydrogen. Allowed photon transitions are those obeying the selection rule l = ±1. The 3p : 2p transition (l = 0) is said to be forbidden, though it may still occur (but only rarely). Fig. 8-8, p.281

Figure 8.9 P(r ) dr is the probability that the electron will be found in the volume of a spherical shell with radius r and thickness dr. The shell volume is just 4r 2 dr. Fig. 8-9, p.282

Figure 8.10 (a) The curve P1s(r) representing the probability of finding the electron as a function of distance from the nucleus in a 1s hydrogen-like state. Note that the probability takes its maximum value when r equals a0/Z. (b) The spherical electron “cloud” for a hydrogen-like 1s state. The shading at every point is proportional to the probability density  1s(r)  2. Fig. 8-10, p.283

4 quantum numbers needed to specify an electron's state The allowed energy levels are quantized much like or particle in a box. Since the energy level decreases a the square of n, these levels get closer together as n gets larger.

Atomic structure

Chemical properties of an atom are determined by the least tightly bound electrons. Factors: Occupancy of subshell Energy separation between the subshell and the next higher subshell.

A pattern starts to emerge s shell l=0 Helium and Neon and Argon are inert…their outer subshell is closed. p shell l=1 Beryllium and magnesium not inert even though their outer subshell is closed…why??