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Please try to include the following

• main points understood, and expand them - what is your understanding of what the points were.
  • expand these points by including many of the details the class discussed.
• main points which are not clear. - describe what you have understood and what the remain questions surrounding the point(s).
  • Other classmates can step in and clarify the points, and expand them.
• How the main points fit with the big picture of QM. Or what is not clear about how today's points fit in in a big picture.
• wonderful tricks which were used in the lecture.
  

First we re-emphasized the difference between bound states and scattering states. From Griffiths we know that we can define bound states as those where the energy is less than zero. Scattering states are those where the energy is greater than zero. To illustrate the difference between bound state and scattering states we examined a few examples with which we are familiar. For each example we determined whether bound states, scattering states, or both are possible. Those examples and their possible states are:

(A) Always bound states : toward x → +/- <math>\infty</math>, E<V no matter how large E is

(B) Always transmission/reflection (In 2D/3D ⇒ scattering)

(C) Depends on

1. Infinite Square Well ⇒ Always Bound States (A)

2. Simple Harmonic Oscillator ⇒ Always Bound States (A)

3. Negative Delta Function (-<math>\alpha</math> *<math>\delta</math>(x)) ⇒(C)

if E<V(+/-<math>\infty</math>)⇒ bound states

if E>V(+/-<math>\infty</math>)⇒ transmission/reflection

4. Positive Delta Function ( <math>\alpha</math> *<math>\delta</math>(x)) ⇒(B) no bound states (E> V(x))

5. Finite Square Well ⇒ (C)

6. Free Particle (B) → wave packet

7. Finite Square Barrier ⇒ Scattering States

*E = 0 – To be honest, I didn't understand this part of the lecture very well.

Whether a particular system can have scattering states, bound states, or both depends on what happens to the potential of the system at x = ±∝. This point is still a little unclear to me. What does the potential do at x = ±∝ for bound states and scattering states? For a system to have bound states the energy of a particle must be less than the potential at some point.

* Next we reviewed what we have covered from CHAPTER 3 so far. Here are the main points we have covered thus far:

– Quantum mechanical operators are Hermitian operators. The eigenvalues of those Hermitian operators are real. (hermitian nature of observable real Eigenvalues exp values)

– Determinate states (Eigenstates) are eigenfunctions of Hermitian operators.

* Next we listed the major topics from chapter 3 that we will cover in the next lecture or so.

– The generalized statistical interpretation of quantum mechanics.

– The uncertainty principle. As it relates to chapter 3.

– Momentum space. Where Ψ(p,t) is used instead of Ψ(x,t).

<math>\ f_p(x) = 1/sqrt(2*pi*hbar)*exp(i*E/hbar*x) </math>

→ normalization ⇒ <math>\int f_p^* (x)*f_p(x) dx</math> = A * <math>\delta</math> (p-p') → <math>\1/sqrt(2*pi*hbar)</math> is corresponded to the value A

Finally, a question was raised concerning notation used in section 3.4 of Griffiths. The question related to equation 3.43 on page 106.

<math> c_n = <f_n|\Psi> </math>.

What is the difference between f_n and Ψ? It should be noted that equation 3.43 is Fourier's trick in bracket notation.

f_n is any function that is any stationary state wave function. You could also think of f_n as the initial state of a wave function.

Ψ is the time-dependent wave function of the system.

<math> \Psi(x,t) = \sum c_n * f_n(x) </math>

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