1. ## Prove closure relations

Let the states ${|n>}$ form a discrete ONB for the space of single particle, and let $\phi_n (\vec{r})$ and $\phi_n (\vec{r}^{'})$ be the wavefunctions for the state ${|n>}$ in the position and wavevector representations, respectively. Prove the so-called closure relations:

$\sum_n {\phi_n^{*}(\vec{r})\phi_n(\vec{r}^{'})=\delta(\ve r-\vec{r}^{'})$ and $\sum_n {\phi_n^{*}(\vec{k})\phi_n(\vec{k}^{'})=\delta(\ve {k}-\vec{k}^{'})$

The second part for k should be the same as the first.

$\sum_n {\phi_n^{*}(\vec{r})\phi_n(\vec{r}^{'})=\sum_n <{\phi_n^{*}(\vec{r})|n>$

I am pretty sure that $\phi_n^{*}(\vec{r})$ can be rewritten as $|\vec{r}>$ which would give

$\sum_n <{\vec{r}|n>$

Since ${|n>}$ form a discrete ONB we should get $\sum_n <{\vec{r}|(\textbf{1}\vec{r}^{'}>)$

I don't know how to justify $\sum_n <{\vec{r}|\vec{r}^{'}>=\delta(\ve r-\vec{r}^{'})$

Or am I totally off in left field again?

Thank you for any help, clarification, or sanity check.

2. ## Re: Prove closure relations

Originally Posted by oldguynewstudent
Let the states ${|n>}$ form a discrete ONB for the space of single particle, and let $\phi_n (\vec{r})$ and $\phi_n (\vec{r}^{'})$ be the wavefunctions for the state ${|n>}$ in the position and wavevector representations, respectively. Prove the so-called closure relations:

$\sum_n {\phi_n^{*}(\vec{r})\phi_n(\vec{r}^{'})=\delta(\ve r-\vec{r}^{'})$ and $\sum_n {\phi_n^{*}(\vec{k})\phi_n(\vec{k}^{'})=\delta(\ve {k}-\vec{k}^{'})$

The second part for k should be the same as the first.

$\sum_n {\phi_n^{*}(\vec{r})\phi_n(\vec{r}^{'})=\sum_n <{\phi_n^{*}(\vec{r})|n>$

I am pretty sure that $\phi_n^{*}(\vec{r})$ can be rewritten as $|\vec{r}>$ which would give

$\sum_n <{\vec{r}|n>$

Since ${|n>}$ form a discrete ONB we should get $\sum_n <{\vec{r}|(\textbf{1}\vec{r}^{'}>)$

I don't know how to justify $\sum_n <{\vec{r}|\vec{r}^{'}>=\delta(\ve r-\vec{r}^{'})$

Or am I totally off in left field again?

Thank you for any help, clarification, or sanity check.
I just realized that this was a Dirac delta function, not the Kronecker delta, so...

$\sum_n <{\vec{r}|n>=<{\vec{r}|\vec{r}^{'}> =\delta(\ve r-\vec{r}^{'})$

Is this the correct conclusion?

Thanks

3. ## Re: Prove closure relations

Hello, we are in the same class, I came up with a proof, and I want to check to make sure it works by asking the professor tomorrow. I will post on here if it doesn't work. The conclusion that <r'|r>=delta(r-r') is correct, but I think your method is wrong as phi_star(r) cannot be written as |r> and must be written as: phi_star(r)=<n|r>. you should be able to re-arrange the functions in the summation at the very beginning. Plug in the relationship I just gave you for the phi and phi_star, and then you will have a SIGMA<r'|n><n|r> which then removing the complete set of states yields <r'|r>.

I apologize for the lack of latex, I'm not sure how to utilize it in the form, I will try a small test in this line. \vec{abc}

4. ## Re: Prove closure relations

If you could tell me how to use latex commands on this site, I will reply to future questions in that fashion(and will probably respond to many of your questions since I am in the same course).

5. ## Re: Prove closure relations

Thanks Zind. To use Latex, there is generally a sigma button on the original post which surrounds the equation with tags "square bracket" TEX "square bracket" then close with "square bracket" /TEX "square bracket"

If you reply with quote, you can copy these tags with control+ C and paste with control+ V.

Look up any TEX on google. Greek letters are \alpha, \beta, etc. Some things you enclose in curly brackets like \sum_n {e^{ikx}} to show what goes with what.

I have done OK on the homework so far but only got a 69 on the first test. I am hoping for a comeback. My undergrad work was at a not so great small college and was 40 years ago, so all this is like I'm seeing it for the first time. Especially all the math, most of the physics concepts I get.

Thanks again. BTW I really like the professor, he is being very patient with me.

6. ## Re: Prove closure relations

Yeah, Dr. Parris is a great guy. I'm in his in person class. I'm in a similar boat HW and test wise (low 70s) it should have been better, except I got hung up on one small part and spent too long on it (excuses excuses.) I am an undergraduate taking the course and ask Dr. Parris many questions, he has also been very patient with me on some of my worst habits. The mathematical basis for a lot of this is sometimes the most difficult part, it's math that has a really high level background (we really only barely touch the surface of it to utilize the results.) But now that I know someone out there is willing to talk about the homwork, I'll check here before the assignments are due to discuss the problems if it's needed! I talked to Dr. Parris, and he encouraged that I do so (he even mentioned that he should start some sort of forum for the class as a whole.)