1. Hydrogen Atom and QM in 3-D 1. HW 8, problem 6.32 and A review of the hydrogen atom 2. Quiz 10.30 3. Topics in this chapter:  The hydrogen atom  The Schrödinger equation in 3-D  The Schrödinger equation for central force Today

2. HW 8, problem 6.32 Following discussion in the textbook, page 196 – 199, to the left of the step ( x < 0 ): Apply smoothness condition: So: To the left of the step ( x < 0 ):  Solve for B and C : with 

3. HW 9, problem 6.32 Conditions given: Because you will have to “tunnel” to Jupiter. Rectangular barrier : The particle (you):  The probability that you end up there is: Which is:

4. The hydrogen atom The electrical potential: To solve the Schrödinger equation in a 3-D polar system is trivial. Let’s start from one of its solutions, the energy level: Example 7.2

5. The Schrödinger equation, from 1-D to 3-D 1-D: 3-D: If choose Cartesian coordinates: Or from:  In a more general case, the coordinate is represented by a vector The Schrödinger equation in 3-D: Now the normalization condition is For bound states, the standing wave is a 3-D standing wave, with energy quantized by 3 quantum numbers, each for one dimension.

6. The 3-D infinite well, just as an example and because Inside the box, if we express the wave function like this:  This leads to three solutions: The Schrödinger equation becomes: And the energy quantization: This is only possible if Here Cartesian coordinates are natural choice, so:  and 

7. Discussion about the energy levels and their wave functions Energy levels: There are three quantum number that define an energy state and its wave function. The ground state: Eave function: Can anyone be 0? Now a special symmetric case, a cube: and Now take for and The energy levels are the same: While the wave functions are not the same: This is called degeneracy. The energy levels are called degenerated.

8. leads to energy levels degeneration. For example, for: Energy levels degenerated and splitting (symmetry broken) The symmetric case: The wave functions are different, but symmetric. Plot the energy levels When the symmetry is broken: for example:  energy level splits

9. Energy levels degenerated and splitting (symmetry broken) splitting Symmetric: Symmetry broken: degenerated Reveals more details

10. The Schrödinger equation for central force Central force: For example: the force: Polar coordinates are a natural choice: with:

11. Solve Schrödinger equation for central force Separate variables: Becomes three equations: radial equation: polar equation: azimuthal equation: Both and are constants. The solution to the azimuthal equation is (the simplest): The z component of the particle’s angular moment is quantized: is called the magnetic quantum number.

12. Where is called the orbital quantum number, and The angular momentum and its quantum numbers Leads to the quantize the magnitude of the particle’s momentum: The solution to the polar equation: Because we have Example 7.3, 7.4 on black board.

13. The shape of an atom of central force The angular probability density for a central force:

14. The radial equation of a central force radial equation: This leads to the solution for energy levels, and the principal quantum number. Degenerate (only depends on n, not l and m l ) Here one needs to know the potential explicitly. Assume hydrogen atom: The relationship of the three quantum numbers (magnetic, orbital and principal): rearrange:

15. The electron “cloud” in the hydrogen atom Electron probability density. Surfaces of constant probability density.

16. Radial probability: the “size” of the atom Radial probability: The Bohr radius: Example 7.6

17. Review questions What are the steps in working out the Schrödinger equation for hydrogen atom? What are the steps in working out the Schrödinger equation for hydrogen atom? How do you connect the quantum numbers introduced in the solutions with those learned from a chemistry class? How do you connect the quantum numbers introduced in the solutions with those learned from a chemistry class?

18. Preview for the next class (10/28) Text to be read: Text to be read: 8.1, 8.2 and 8.3 8.1, 8.2 and 8.3 Questions: Questions: What had the Stern-Gerlach experiment been designed to prove? What it actually proved? What had the Stern-Gerlach experiment been designed to prove? What it actually proved?

19. Homework 11, due by 11/6 Problems 7.37, 7.38 and 7.45 on page 281.