Responsible party: liux0756, Dagny

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Quiz 2 main concepts: quiz_2_1023

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.

Important Info regarding the quiz:

• Quiz 2 is on Friday October 23, 2009 (in our lecture room at our lecture time)
• Quiz 2 covers material subsequent to Quiz 1 and through Section 3.3.1
• A practice quiz is posted. See Yuichi for questions regarding this practice quiz.

Major topics:

• Free particle
• Finite Square Well
• Bound States and Scattering States
• Delta function Potential
• Hilbert Space
• Inner profuct of two functions
• Hermitian Operators
• Eigenfunctions

We reviewed a little from the chapter 3 material. We had several questions and spent some time on them.

Highlights from Chapter 3:

Part 1. Inner Product

• The inner product of two functions:

“<math>\int_a^b f(x)*g(x)\,\mathrm dx = <f|g></math>”

• <math>|f></math> and <math>|g></math> are vectors in Hilbert space. <math>|f>=\begin{bmatrix}f_1\\f_2\\f_3\\.\\.\\.\end{bmatrix}</math>, <math>|g>=\begin{bmatrix}g_1\\g_2\\g_3\\.\\.\\.\end{bmatrix}</math>. Then dot product <math><f|g>=[f_1^* f_2^* f_3^* …]\begin{bmatrix}g_1\\g_2\\g_3\\.\\.\\.\end{bmatrix}=\sum_{i=1}^\infty f_i^*g_i</math>.

• unit vectors <math>|e_i></math> are orthonormal, <math><e_i|e_j>=\delta_i_j</math>

• <math>\sum|e_i><e_i|=\begin{bmatrix}

1 & 0 & 0 & …
0 & 1 & 0 & …
0 & 0 & 1 & …
. & . & .
. & . & .
. & . & . \end{bmatrix}=I</math>, unit matrix.

• unit vector can be used to help calculating inner product:

<math><f|g>=\sum_{i} <f|e_i><e_i|g></math>, as <math><f|e_i>=f_i^*</math>, <math><e_i|g>=g_i</math>, so <math><f|g>=\sum_{i} f_i^*g_i</math>

• <math><e_i|g>=g_i</math> means the projection of vector <math>|g></math> to <math>|e_i></math> basis.

• operator <math>\hat{Q}=\begin{bmatrix}

Q_1_1 & Q_1_2 & Q_1_3 & …
Q_2_1 & Q_2_2 & Q_2_3 & …
Q_3_1 & Q_3_2 & Q_3_3 & …
. & . & .
. & . & .
. & . & . \end{bmatrix}</math> <math><e_i|\hat{Q}|e_j>=Q_i_j</math> <math><f|\hat{Q}|g>=\sum_{i,j} <f|e_i><e_i|\hat{Q}|e_j><e_j|g> = \sum_{i,j} f_i^* Q_i_j g_j </math>

Part 2. Hermitian Operator

• Hermitian operators are operators such that:

<f|Âg> = <Âf|g> for all f(x) and all g(x)

where  is a hermitian operator and f and g are functions of x.

Hermitian Operators represent observables.

Âf = af

Eigenvalues are numbers only. (i.e., they are NOT operators or functions)

• Eigenvalue equation for operator Â, where f is the eigenfunction and a is the eigenvalue.

• In any states, the eigenvalues of a Hermitian operator are all real numbers.

• If the eigenvalues of an operator are real in any states, the operator must be a Hermitian operator.

• The mechanical variables which can be observed in experiments require their expectation values to be real numbers. So the corresponding operators must be Hermitian operators.

To be continued :)

Good luck on the quiz!

To go back to the lecture note list, click lec_notes
previous lecture note: lec_notes_1019
next lecture note: lec_notes_1026
Quiz 2 main concepts: quiz_2_1023