1. Bound States 1. A quick review on the chapters 2 to 5. 2. Quiz 10.9. 3. Topics in Bound States:  The Schrödinger equation.  Stationary States.  Physical conditions: well-behaved functions.  A review of classical bound states.  The infinite potential well.  The finite potential well.  The simple harmonic oscillation.  Expectation values, uncertainties, and operators. today

2. Review Large Small Slow Fast Classical physics Special Relativity Quantum mechanics smooth wave function EM wave  particle S.R. particle  wave Schrödinger Equation What is here? Relativistic quantum mechanics Quantum electrodynamics, …

3. Review of Special Relativity S and S’ system: S and S’ system: For a particle with velocity in S: For a particle with velocity in S: The Doppler effect: The Doppler effect: S’ moves with velocity v in S along the x-axis. When =0, the course is moving away from the observer. When θ =0, the course is moving away from the observer.

4. Review of EM wave  particle EM wave behave as particle: EM wave behave as particle: Proof: Proof: Blackbody radiation. Blackbody radiation. Photoelectric effect. Photoelectric effect. Compton Scattering. Compton Scattering. Energy (EM wave) converts into matter (particle) Energy (EM wave) converts into matter (particle) Pair production. Pair production.

5. Review: matter waves and the free particle Schrödinger equation The de Broglie wavelength of a particle: The de Broglie wavelength of a particle: The frequency: The frequency: The h-bar constant: The h-bar constant: The connection between particle and wave: The connection between particle and wave:  momentum  energy Wave number and angular frequency: Wave number and angular frequency: The free particle Schrödinger equation: The free particle Schrödinger equation: And the plane wave solution: And the plane wave solution: The Heisenberg Uncertainty Principle The Heisenberg Uncertainty Principle

6. The Schrödinger equation From the free particle Schrödinger Equation We understand this equation as energy accounting  So the time dependent Schrödinger Equation And its plane wave solution And this leads to the equation that adds an external potential Solve for with the knowledge of, for problems in QM.

7. Solve the Schrödinger equation When the wave function can be expressed as the time independent Schrödinger Equation We have The solution of this equation is the stationary states because The probability of finding a particle does not depend on time: and Normalization : The wave function be smooth  the continuity of the wave function and its first order derivative. Two conditions

8. Solving the Schrödinger equation. Case 1: The infinite potential well Equation and Solution: Energy and probability 1.Standing wave. 2.The QM ground- state. A bound state particle cannot be stationary, although its wave function is stationary. 3.Energy ratio at each level: n 2. 4.With very large n, QM  CM. Examples 5.1, 5.2

9. Solving the Schrödinger equation. Case 2: The infinite potential well The change  Equations:

10. Solving the Schrödinger equation. Case 2: The finite potential well What is the idea? How would you verify this? solutions Energy quantization Penetration depth Penetrating into classically forbidden regions depth

11. Compare Case 1 and Case 2: The infinite and finite potential wells Finite well

12. Solving the Schrödinger equation. Case 3: The simple harmonic oscillator This model is a good approximation of particles oscillate about an equilibrium position, like the bond between two atoms in a di-atomic molecule. Example 5.3 Solve for wave function and energy level

13. Solving the Schrödinger equation. Case 3: The simple harmonic oscillator Energy are equally spaced, characteristic of an oscillator Wave function at each energy level Gaussian

14. Expectation values, uncertainties, and operators expectation value of x uncertainty operator  Probability weighted average Probability weighted average of x 2 is

15. Expectation values, uncertainties, and operators expectation value (observables) uncertainty Basic operator Functions of operator So

16. Review questions Follow the definition of operator and expectation value, how do you understand the Schrödinger Equation now? Follow the definition of operator and expectation value, how do you understand the Schrödinger Equation now? How do you now understand the tunneling effect mentioned often in semiconductor fabrication industry? How do you now understand the tunneling effect mentioned often in semiconductor fabrication industry? For operator, how do you calculate its expectation value and uncertainty? For operator, how do you calculate its expectation value and uncertainty? Please summarize this chapter yourself. Please summarize this chapter yourself.

17. Preview for the next class (10/21) Text to be read: Text to be read: In chapter 6: In chapter 6: Section 6.1 Section 6.1 Section 6.2 Section 6.2 Section 6.3 Section 6.3 Section 6.4 Section 6.4 Questions: Questions: Have you heard of the “tunneling effect” in the EE department (only for EE students)? Have you heard of the “tunneling effect” in the EE department (only for EE students)? What is a wave phase velocity? What is a wave group velocity? What is a wave phase velocity? What is a wave group velocity?

18. Homework 7, due by 10/16 1. Problem 5 on page 186. 2. Problem 24 on page 187. 3. Problem 28 on page 187. 4. Problem 34 on page 188.