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.

• The product of two Hermitian operators is a Hermitian operator only if the two Hermitian operators are commutable (which means [A,B]=AB-BA=0)

proof: A, B are two Hermitian operators, so <math>A^+=A</math>, <math>B^+=B</math> (see Problem 3.5)

then <math>(AB)^+=B^+A^+=BA</math> (see Problem 3.5)

Only if BA=AB can we get that <math>(AB)^+=BA=AB</math>. AB is Hermitian. QED

Part 3. Orthonormality for eigenstates

• We can verify that for infinite square well and simple harmonic oscillator, the eigenstates are orthonormal. (This is not proved in class, but if you are interested, you can see it in the end of this lecture notes)

This can be generalized to all Hermitian operators. (Theorem 2 in textbook) Eigenfunctions belonging to distinct eigenvalues are orthogonal.

<math><\psi_n|\psi_m>=<n|m>=\int \psi_n^* \psi_m\, dx =\delta_n_m</math>

• For degenerate states, we can use the Gram-Schmidt orthogonalization procedure to construct orthogonal eigenfunctions within each degenerate subspace.

• Any state can be expressed by linear combination of eigenstates.

<math>|\psi>=\sum C_n |\psi_n> </math>

<math>|C_n|^2</math> is the probability that the particle is found to have energy <math>E=E_n</math>

• <math>\sum |C_n|^2 = 1 </math>

proof: because of orthonormality, <math><\psi|\psi>=1</math>, <math><\psi_n|\psi_m>=\delta_n_m</math>

so <math>1=<\psi|\psi>=\sum_{n} \sum_{m} C_n^* C_m <\psi_n|\psi_m>=\sum_{n} \sum_{m} C_n^* C_m \delta_n_m=\sum_{n} |C_n|^2 </math>. QED

• The expectation value of energy can be expressed as <math><\hat{H}>=\sum_{n} |C_n|^2 E_n </math>

proof: in eigenstates, the expectation values are eigenvalues, which means <math><\hat{H}|\psi_n>=E_n|\psi_n></math>

so <math><\psi_n|\hat{H}|\psi_m>=<\psi_n|E_m|\psi_m>=E_m<\psi_n|\psi_m>=E_m \delta_n_m </math>

The expectation value of H in state <math>\psi</math> is: <math><\hat{H}>=<\psi|\hat{H}|\psi>=\sum_{n} \sum_{m} C_n^* C_m <\psi_n|\hat{H}|\psi_m> = \sum_{n} \sum_{m} C_n^* C_m E_m \delta_n_m = \sum_{n} |C_n|^2 E_n </math> QED.

Proof of orthonormality of infinite square well and simple harmonic oscillator(this part is not mentioned in class)

• infinite square well

The wave function is <math>\psi_n(x)=\sqrt{\frac{2}{L}} \sin\frac{n \pi x}{L} </math>

So <math> <\psi_n|\psi_m>=\frac{2}{L}\int_0^L \sin\frac{n \pi x}{L} \sin\frac{m \pi x}{L} dx

=\frac{1}{L}\int_0^L [\cos\frac{(n-m) \pi x}{L} - \cos\frac{(n+m) \pi x}{L}] dx </math>

If n=m, then <math> <\psi_n|\psi_m>=<\psi_n|\psi_n>=\frac{1}{L}\int_0^L [1 - \cos\frac{2n \pi x}{L}] dx

=\frac{1}{L}(L - \frac{L}{2n \pi} \sin\frac{2n \pi x}{L} |_0^L ) = \frac{1}{L} (L-0)=1</math>

If <math>n \neq m </math>, then <math> <\psi_n|\psi_m>=\frac{1}{L}[\frac{L}{(n-m) \pi} \sin\frac{(n-m) \pi x}{L} |_0^L - \frac{L}{(n+m) \pi} \sin\frac{(n+m) \pi x}{L} |_0^L ]=\frac{1}{L}(0-0)=0 </math>

So <math> <\psi_n|\psi_m>=\delta_n_m</math> QED.

• simple harmonic oscillator

The wave function is <math>\psi_n(x)=(\frac{mw}{\pi \hbar})^{1/4} \frac{1}{sqrt{2^n n!}} H_n(\xi)e^{-\xi^2 /2} </math> (Equation [2.85] in textbook)

So <math> <\psi_m|\psi_n>=sqrt(\frac{mw}{\pi \hbar}) \frac{1}{sqrt{2^m m!}} \frac{1}{sqrt{2^n n!}} \int_{-\infty}^{+\infty} H_m(\xi)H_n(\xi)e^{-\xi^2} dx </math>

note that <math>\xi=sqrt(\frac{mw}{\hbar})x</math> (equation [2.71] in textbook), we have <math>d\xi=sqrt(\frac{mw}{\hbar})dx</math>, then <math> <\psi_m|\psi_n>=\frac{1}{sqrt{\pi}} \frac{1}{sqrt{2^m m!}} \frac{1}{sqrt{2^n n!}} \int_{-\infty}^{+\infty} H_m(\xi)H_n(\xi)e^{-\xi^2} d\xi </math>

According to equation [2.89] in textbook, we have

<math>e^{-t^2+2t \xi}=\sum_{m=0}^{\infty} \frac{t^m}{m!} H_m(\xi)</math>

<math>e^{-s^2+2s \xi}=\sum_{n=0}^{\infty} \frac{s^n}{n!} H_n(\xi)</math>

multiply the two equations above we get

<math>e^{-(t^2+s^2)+2(s+t) \xi}=e^{2ts+\xi^2}e^{-(t+s-\xi)^2}=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} \frac{t^m s^n}{m!n!} H_m(\xi) H_n(\xi)</math>

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