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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.
  

Define Hilbert space- Hilbert space contains the set of all square-integrable functions, on a specified interval (usually ±∞, but more generally (a and b)
<math>f(x)=\int_{-\infty}^{\infty}|f(x)|^2dx < \infty</math>
where all functions that have this property make up a vector space we call Hilbert space. Hilbert space is real space that contains the inner product of two functions defined as follows:
<math> <f|g>=\int_{-\infty}^{\infty}f(x)^*g(x) dx </math>if f and g are both square-integrable the inner product is guaranteed to exist, if the inner product does not exist the then <math> <f|g>=\int_{-\infty}^{\infty}f(x)^*g(x) dx </math> diverges
*Notation relationships
We claim that the operator <math>\hat{Q}\</math>⇔<math>Q_{mn}=\int\psi_m^*\hat{Q}\psi_n dx</math>. Here we introduce new short hand notation <e_m|<math>\hat\</math>{Q}e_n>, where |e_n> is a unit vector. For example |e_n> could represent the ground state wave function.
Back to the claim made. Is this a sensible claim?

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