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Eigenvalues of an Operator

  1. Nov 27, 2007 #1
    If I have an operator of the form [tex]1+3\vec{e}\cdot\vec{\sigma}[/tex] where [tex]\vec{e}\cdot\vec{e}=1[/tex].

    How can I find the eigenvalues quickly?
  2. jcsd
  3. Nov 27, 2007 #2


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    you write out the matrix representation of the operator, and then you find the eigenvalues and eigenvectors to that matrix, same as you do in linear algebra.
  4. Nov 27, 2007 #3
    How can I write out the matrix representation without knowing e?
  5. Nov 27, 2007 #4


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    You know this:


    Why not just make the ansatz:
    [tex] \vec{e} = (a,b,c) [/tex]
    [tex] a^2 + b^2 + c^2 = 1 [/tex]

    When you dont have any numbers or explicit expressions, but you have a condition to be fulfilled, you can atleast do an asatz.
  6. Nov 27, 2007 #5
  7. Nov 27, 2007 #6


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    Or, you could just choose to use a coord system in which e is in the z direction.
  8. Nov 28, 2007 #7
    listen man if you download the book Schaum's Outline of Quantum Mechanics off emule, go to page 54 there's a whole section on how to represent an operator in matrix form. There are also plenty of problems on the subject in 5,6,7.
    I guess you can also check these stuff in the Cohen-Tannoudji book, also availabe in emule. Good luck.
    And by the way, I find those one line advices to be very unhelpful. that's why I usually turn to the books.
  9. Nov 29, 2007 #8


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    Is it a legal download?

    why download a book that costs 12$ ?
  10. Dec 4, 2007 #9
    Is it not a legal download. However, if this book was available on the internet as a scanned and well edited pdf file, I'd be more than glad to pay for it as this price. Just as I used to illegaly download mp3 before the age of iTunes.
    Furthermore, as an undergraduate myself, I feel the moral need to help other undergrads regardless of who they are and where they live.
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