Results 1 to 6 of 6
Like Tree1Thanks
  • 1 Post By Zind

Thread: Prove closure relations

  1. #1
    Senior Member oldguynewstudent's Avatar
    Joined
    Oct 2009
    From
    St. Louis Area
    Posts
    255

    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><n|\phi_n(\vec{r}^{'})>

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

    \sum_n <{\vec{r}|n><n|\vec{r}^{'}>

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member oldguynewstudent's Avatar
    Joined
    Oct 2009
    From
    St. Louis Area
    Posts
    255

    Re: Prove closure relations

    Quote Originally Posted by oldguynewstudent View Post
    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><n|\phi_n(\vec{r}^{'})>

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

    \sum_n <{\vec{r}|n><n|\vec{r}^{'}>

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

    Is this the correct conclusion?

    Thanks
    Last edited by oldguynewstudent; Mar 16th 2015 at 05:07 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2015
    From
    United States
    Posts
    3
    Thanks
    1

    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}
    Thanks from oldguynewstudent
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Mar 2015
    From
    United States
    Posts
    3
    Thanks
    1

    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).
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member oldguynewstudent's Avatar
    Joined
    Oct 2009
    From
    St. Louis Area
    Posts
    255

    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.
    Last edited by oldguynewstudent; Mar 17th 2015 at 02:57 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Mar 2015
    From
    United States
    Posts
    3
    Thanks
    1

    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.)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. what is the symmetric closure of below relations?
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Jan 27th 2015, 02:21 PM
  2. Prove a transitive closure
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Oct 30th 2012, 07:12 AM
  3. Closure relations on a language?
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: Sep 8th 2011, 12:27 PM
  4. Determining transitive closure and prove
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: Aug 27th 2009, 09:01 AM
  5. Replies: 6
    Last Post: Feb 11th 2009, 12:56 PM

Search Tags


/mathhelpforum @mathhelpforum