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Relationship between several operators and their eigenvectors.

  1. Nov 28, 2012 #1
    1. The problem statement, all variables and given/known data
    operators: K=LM and [L,M]=1

    α is an eigenvector of K with eigenvalue λ.

    Show that x=Lα and y=Mα are also eigenvectors of K and also find their eigenvalues.


    2. Relevant equations
    K=LM
    [L,M]=1
    Kα=λα


    3. The attempt at a solution
    I tried, but its not even worth putting up here.
     
  2. jcsd
  3. Nov 28, 2012 #2

    mfb

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    2016 Award

    Staff: Mentor

    You have to show that x eigenvector of K which means Kx=λ'x for some λ'.

    Kx=KLα = LMLα
    Try to modify that equation (using [L,M]=1) until you can use Kα=λα and simplify.
     
  4. Nov 28, 2012 #3
    So, am I doing this right?
    Starting with: Kx=λ'x where x=Lα

    Kx=KLα=LMLα ===> using K=LM

    =L(LM-1)α=LLMα-Lα ===>using [L,M]=1

    =LKα-Lα
    Now going back to Kx=λ'x
    LKα-Lα=λ'Lα

    LKα=(λ'+1)Lα

    Finally,

    Kα=L-1(λ'+1)Lα

    since λ'+1 is a constant,

    Kα=(λ'+1)L-1
    Kα=(λ'+1)α

    so λ'=λ-1.

    is that correct?
     
    Last edited: Nov 28, 2012
  5. Nov 28, 2012 #4
    Good start! And correct finish!

    But I would have continued like this:

    = Lλα - Lα since Kα = λα.

    = (λ - 1)Lα since λ commutes with the operator L.

    Hence K(Lα) = (λ - 1)Lα, showing that Lα is an eigenvector of operator K with eigenvalue (λ - 1).
     
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