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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.

*Special Functions *Raising/Lowering Operators *Structuring DEs *Expectation of Momentum / Momentum Squared *Orthogonality of Hermitian Operators *Introduction to the Analytical Method

One of the first points made today was that special functions such as the Bessel Functions, Legendre Polynomials and Hermite Polynomials keep emerging as solutions to various classes of differential equations. The important aspect of this fact is that when one function of a class is a solution, usually other functions in that class and sometimes linear combinations of the functions are solutions as well. One of the most familiar examples is the types of differential equations we commonly see as physics students: <math> y + k^2 y = 0</math>, <math> y +b y' k^2 y = 0</math>, etc., which have exponential solutions. Similar DEs have other exponential solutions, and using Euler's Formulae we can express these as trigonometric functions; all being a class of exponential functions that form solutions to these types of differential equations. This concept pertains to Quantum Mechanics with the exponential and trig solutions we've used thus far, and with the Hermite Polynomials in the Analytical solution for the wave functions of the Harmonic Oscillator (see pg 56), which we'll cover Wednesday.

With these, the Hamiltonian can be written as <math> H=\hbar\omega(a_+a_- + 1/2) </math>.

Which makes the Schrodinger equation <math> \hbar\omega(a_\mp a_\pm \pm (1/2))\psi=E\psi </math>

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