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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.

• Reiterating and justification for why <math> \frac{d}{dt} \int f(x)dx=\int \frac{\partial}{\partial t}f(x)dx=0</math> which extends specifically to our application: <math> \frac{d}{dt} \int_{-\infty}^{\infty}\Psi^*\Psi dx=\int_{-\infty}^{\infty}\frac{\partial}{\partial t}(\Psi^*\Psi)dx=0</math>

• Waveforms, Fourier Transform Analysis

CHAPTER 2 Today's Main points:

• Energy Quantization
• Time Independent Schrodinger equation
• Method of Separation of Variables / class of separable solutions
• Simple Harmonic Oscillator - (We'll cover this in another lecture soon)
• Stationary States
• The Free Particle
• Infinite and Finite Square Wells - (We'll go more in depth soon)

• When a particle is bound by a potential, V, and the total energy, E, is less than V: E < V, there are specific allowed energy levels that a particle can have. A particle can have any of the allowed energies, but not an energy level that is not allowed.
• The Infinite Square Well is an example of a bound state with quantized energy (Griffiths p30-38)
• The Finite Square Well is another bound state with quantized energy (Griffiths p78-82)
• Quantized Energy states do not exist for the ideal Free Particle (Griffiths p59-67) which may have any of a continuous range of energies.

Separable Solutions are a very distinct class of solutions which may be broken down into products of each variable: <math>f(x,t)=g(x)h(t)</math>. Physically these solutions represent a special case and therefore a very small portion of the number of potential solutions that may not be separated into product functions. Mathematically these product solutions can be solved relatively easily with purely analytical theory, the method of Separation of Variables.

The Method of Separation of Variables takes advantage of cases of separable solutions. Derived in Griffiths p24-28, we can separate <math>\Psi(x,t)</math> into two product functions <math>\psi(x)*\phi(t)</math>. With a product solution, we can rearrange and substitute so the Schrodinger equation reads <math>i\hbar\frac{1}{\phi}\frac{d\phi}{dt}=-\frac{\hbar^2}{2m}\frac{1}{\psi}\frac{d^2\psi}{dx^2}+V</math> The key here is that the left side depends only on t and the right side depends only on x. You could vary either t or x and fix the other, and the equation must still be satisfied. This can only be true if both sides are equal to a constant, and furthermore the same constant.

1. -Keep in mind that this only works for separable solutions. That is, solutions of the Schrodinger Equation that can be separated in to a product of two functions, each of which only depends on one variable. This is a narrow class of solutions, and potentially very few of all of the solutions that exist would satisfy these conditions so it shouldn't surprise you that this analysis seems to be valid only for a very special case.

If each side of the above separated Schrodinger equation is equal to a constant, E, we can write the time-dependent equation as: <math>\frac{d\phi}{dt}=-\frac{iE}{\hbar}\phi </math> which has the easily obtained exponential solution: <math> \phi(t)=e^{-iEt/\hbar} </math>

<math>-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V\psi = E \psi </math>