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

2. 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

3. 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

4. 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

5. Table 8-1, p.265

6. 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

7. 28.14 MeV
4.05 MeV
Figure (a)  decay of a radioactive nucleus. (b) The potential energy seen by an  particle emitted with energy E. R is the nuclear radius, about 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

8. 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

9. 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

10. Table 8-2, p.269

11. Table 8-3, p.269

12. Table 8-4, p.280

13. Table 8-5, p.280

14. 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

15. 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

16. 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

17. 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.

18. Atomic structure

20. 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.

21. 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??