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Math Help - QM - Base Kets and Matrix Representations Part 1

  1. #1
    Senior Member bugatti79's Avatar
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    QM - Base Kets and Matrix Representations Part 1

    FOlks,

    I am having difficulty understanding from image 144.jpg eqn 1.3.25, how Sakurai identified this as a square matrix by column matrix.

    Can anyone shed light on this? THanks
    Attached Thumbnails Attached Thumbnails QM - Base Kets and Matrix Representations Part 1-imag0145.jpg   QM - Base Kets and Matrix Representations Part 1-imag0144.jpg   QM - Base Kets and Matrix Representations Part 1-imag0143.jpg  
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  2. #2
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    Well, I wouldn't use the language "identified this as a square matrix by column matrix." Sakurai appears to have identified Eq. 1.3.25 as an application of the rule for multiplication of a column vector by a square matrix. The final result, as we can see, is a bracket (a scalar). To go from the first line of 1.3.25 to the second, all Sakurai did was insert the identity in the form of

    \displaystyle I=\sum_{a''}|a''\rangle\langle a''|.

    That's certainly a square matrix.

    I guess I'm not entirely sure what your question is. Perhaps my comments could help to clarify your question? Are you asking why Sakurai can write down Eq. 1.3.25? Or are you asking how he recognized 1.3.25 as, partly, the multiplication of a square matrix with a column vector? Or something else?
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  3. #3
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Ackbeet View Post
    Or are you asking how he recognized 1.3.25 as, partly, the multiplication of a square matrix with a column vector? Or something else?
    Apologies for lack of clarity...yes this what I am looking for...additionally how he arrives on the next line 1.3.26? Ie how the kets are column matrices and the bra as row matrices? I am just looking for that link...

    Thanks
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    Take a look at this thread. That might help explain a few things. Let me know if you have other questions.
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  5. #5
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Ackbeet View Post
    Take a look at this thread. That might help explain a few things. Let me know if you have other questions.
    THis an interesting thread particularly

    Quote Originally Posted by Ackbeet View Post
    Ok. Let's start with the real case. The basic idea here is to write the inner product in terms of matrix multiplication. If I have vector \mathbf{x} and vector \mathbf{y}, and I want to form their inner product, I do this:

    \displaystyle\mathbf{x}\cdot\mathbf{y}=\sum_{j=1}^  {n}x_{j}y_{j}, right?

    Now, if \mathbf{x} and \mathbf{y} are both column vectors, I could write this using matrix multiplication as follows:

    \mathbf{x}^{T}\mathbf{y}.

    In this expression, the usual matrix multiplication does for me what the sum does in the previous expression. With me so far?


    You say that x and y are column vectors, ie the kets alpha and gamma are column vectors in eqn 1.3.26.

    But I still dont see how these arrive to be column matrices...so I guess will have to take it as given for the moment :-)

    Pardon my ignorance on this!
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    Typically, kets are defined as column vectors. (Incidentally, in your last post, you use the phrases "column vector" and "column matrix". Are you meaning the same thing by those two terms?) Bras are something for which you have to deduce their structure such that the inner product makes sense - they are vectors in the dual space.
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    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Ackbeet View Post
    Typically, kets are defined as column vectors. (Incidentally, in your last post, you use the phrases "column vector" and "column matrix". Are you meaning the same thing by those two terms?) Bras are something for which you have to deduce their structure such that the inner product makes sense - they are vectors in the dual space.
    Ok, that is good information. I also see that an inner product is a complex number therefore in matrix representation this has to be in the form

    (1*N)(N*N)(N*1) =(1*1) to get a number and this equivalent to

     <br />
\langle a|\hat X|b \rangle<br />

    Additionally

    (1*N)(N*1)=(1*1) equivalent to \langle a| b \rangle

    (N*1)(1*N)=(N*N) equivalent to \langle b| a \rangle

    Hope I have them right :-)
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    Quote Originally Posted by bugatti79 View Post
    Ok, that is good information. I also see that an inner product is a complex number therefore in matrix representation this has to be in the form

    (1*N)(N*N)(N*1) =(1*1) to get a number and this equivalent to

     <br />
\langle a|\hat X|b \rangle<br />

    Additionally

    (1*N)(N*1)=(1*1) equivalent to \langle a| b \rangle

    (N*1)(1*N)=(N*N) equivalent to \langle b| a \rangle

    Hope I have them right :-)
    The last one there should be |b\rangle\langle a|, but yes, that's the idea.
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  9. #9
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Ackbeet View Post
    The last one there should be |b\rangle\langle a|, but yes, that's the idea.
    Yes your right, thats a typo.

    Thanks alot :-)
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  10. #10
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    You're welcome! Keep the QM coming - it's loads of fun for me. *grin*
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  11. #11
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Ackbeet View Post
    You're welcome! Keep the QM coming - it's loads of fun for me. *grin*
    Ok...with pleasure
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