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Main class wiki page: home

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.

• We mostly were learning how to solve schrodinger's time independent equation given a potential.
• we learned that we can have either bound states or scattering depending on the energy of the particle compared with the potential.
• for the bound state the main thing physicists care about is the fact that the energy is quantized.

• we can represent a function by a vector.

<math>\ f(x) = \sum_n c_n\psi_n</math> can be written as <math>\ f(x) ⇔\normalsize \left(\large\begin{array}{GC+23} \\c_{1}\\c_{2}\\c_{n}\end{array}\right)\ {\Large</math> where <math>\ c_n = \int f(x)*\psi_n(x) \,\mathrm dx</math>

• the above vector can have finite or infinite number of components depending on how many stationary states there are.
• in the language of vectors, operators are transformations and are featured by Matrices. (I think one point of the lecture was the following: by operation on a function you get another function analogous to for example when you apply addition rules on a vector you get another vector which belongs to the original vector space)
• when we operate on a stationary wave-function we get the wave-function multiplied by a constant.

<math>\hat{H}\psi_n(x)=E\psi_n(x),</math>

• in general when we have an arbitrarily wave-function f(x), we get:

<math> \hat{H} f(x)=\hat{H} \sum_n c_n\psi_n(x)=\sum_n c_n E_n\psi_n⇔\normalsize \left(\large\begin{array}{GC+23} \\c_{1}E_{1}\\c_{2}E_{2}\\c_{n}E_{n}\end{array}\right)\ {\Large</math> where <math> \sum_n c_n E_n</math> is a constant.

• observable quantities like position, momentum, energy and etc. are represented by operators.
• operators for physically observable quantities are Hermitians.

if the following quality holds, then the operator is a Herimitian <math>\int \psi(x)*Q\psi(x)\,\mathrm dx = \int (Q\psi(x))*\psi(x)\,\mathrm dx </math>.

• average value of a Hermitian operator is always real.once you assume this you can show that the operator is Hermitian meaning that you can show the above equality holds.
• physicist's notation for a dot product is <math>\ <\psi|Q\psi> </math> where <math>\ \psi </math> is a vector and <math>\ Q\psi </math> is another vector.
• since in quantum mechanics we usually are dealing with complex quantities, we might change the value of a dot product if we change the order of the dot product. if the quantity is complex, we get its complex conjugate back,

<math>\ <\psi|Q\psi> = <Q\psi|\psi>* ,</math> and if it is real we are not changing anything by changing the order.

• determinate state is a state with no uncertainty and each measurement of some quantity Q gives bak the same result q.

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