1. Bound States
Review of chapter 4. Comment on my errors in the lecture notes.
Quiz 9.30.
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: matter waves and the free particle Schrödinger equation
The de Broglie wavelength of a particle:
The frequency:
The h-bar constant:
The connection between particle and wave:
 momentum
 energy
Wave number and angular frequency:
The free particle Schrödinger equation:
And the plane wave solution:
The Heisenberg Uncertainty Principle

3. The Schrödinger Equation
Bound state: particle’s motion is restricted by external force to a finite region of space.
Free particle
With external potential
Bound state
Unbound state
Bound state
ideal spring
Unbound state
the spring breaks

4. The Schrödinger Equation
Free particle
Plane wave solution leads to the energy accounting
With external potential U(x)
So the time dependent Schrödinger Equation 

5. Quantum Mechanics – problem solved
In classical mechanics
Solve for with the knowledge of
In quantum mechanics
Solve for with the knowledge of

6. The stationary states and the time independent Schrödinger Equation
Exam the case
In English: time and space can be separated in a wave function
The temporal part 
From   
the time independent Schrödinger Equation
The spatial part
The stationary states: probability of finding a particle does not depend on time:

7. Physical conditions: well-behaved functions
Mathematically:
Normalization:
Physically: the wave function must be smooth, requiring the continuity of the wave function and its first order derivative.
Both conditions will be used when solving the Schrödinger equation for the wave function.

8. A review of classical bound states
Particle confined to
Classically forbidden region exists.

9. Atomic bound states When
Atom 2 is bound to the potential well with Atom 1.
But this bound is not the classical sense.

10. Review questions
Follow the free particle Schrödinger Equation, with the energy accounting argument based on the plane wave solution, can you “derive” the time dependent Schrödinger Equation?
With the assumption of a time and space separated wave function, can you “derive” the time independent Schrödinger Equation? Why in this equation we replace the partial derivative with normal derivative?
In an infinitely deep well with vertical walls, and assuming elastic scattering, can you describe (words, or equations) a particle with KE inside this well?

11. Preview for the next class
Text to be read:
In chapter 5:
Section 5.5
Section 5.6
Section 5.7
Section 5.8
Questions:
Why do we arrive at standing wave solution for particles in infinity well?
Have you heard of the “tunneling effect” in the EE department (only for EE students)?
Mechanical standing waves can be thought of two waves passing each other. (For ME students) can you describe this mathematically?