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Eigenvalues, eigenvectors, eigenstates and operators

  1. May 11, 2013 #1
    1. The problem statement, all variables and given/known data

    Good evening :-)

    I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could point me in the right direction:

    S2) Show that the state vectors |Sx+> = [itex]\frac{1}{\sqrt{}2}[/itex] times a 2x1 matrix (1,1) and |Sy+> = [itex]\frac{1}{\sqrt{}2}[/itex] times a 2x1 matrix (1,-1) are eigenvectors of Sx = h/2 times a 2x2 matrix (0 1, 1 0) with respective eigenvalues plus and minus h/2...


    Part two... Of what operator is the state [itex]\frac{1}{\sqrt{}2}[/itex](|Sx+> + |Sy+>) and eigenstate, and with what eigenvalue...

    Any help would be great and much appreciated

    2. Relevant equations

    All in question

    3. The attempt at a solution

    Part 1: This I can do by using |A - λI| = 0, finding the eigenvalues, then using A.v=λv and setting up simutaneous quations to find the eigenvalues.

    Part 2... this is where I need help please :-)
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 11, 2013 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Hello.

    For part (1) you don't need to go through solving for the eigenvalues and eigenvectors. You just want to verify that |Sx+>, say, is an eigenvector of the given matrix. So, just multiply the matrix times the 2x1 vector representing |Sx+> and verify that you get a constant factor times |Sx+>. The constant factor is your eigenvalue.

    For part (2), what 2x1 vector do you get when you add [itex]\frac{1}{\sqrt{}2}[/itex](|Sx+> + |Sy+>) ? Can you recognize it?

    [Aside: The notation|Sy+> for the second given vector is a bit odd. It seems like |Sx-> would be more appropriate.]
     
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