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Determining eigenvalue problem

  1. Sep 8, 2009 #1
    I'm trying to teach myself quantum mechanics using a book I got. I made an attempt at one of the questions but there are no solutions or worked examples so I'm wondering if I got it right.

    Here it goes

    1. The problem statement, all variables and given/known data
    Suppose an observable quantity corresponds to the operator [tex]\hat{B}= -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}[/tex].

    For a particular system, the eigenstates of this operator are
    [tex]\Psi(x)=Asin\frac{n\pi x}{L}[/tex], where n = 1,2,3,...; A is the normalisation constant

    Determine the eigenvalues of [tex]\hat{B}[/tex] for this case




    2. Relevant equations

    [tex]\hat{A}\psi_{j}=a_{j}\psi_{j}[/tex] I think


    3. The attempt at a solution
    I used the operator on [tex]\psi[/tex] and differenciated twice to get
    [tex]\frac{\hbar^2 n^2 \pi^2}{2mL^2}ASin\frac{n\pi x}{L}[/tex]
    this corresponds to [tex]a_j\psi_j[/tex] so my answer for the eigenvalues is

    [tex] \frac{\hbar^2 n^2 \pi^2}{2mL^2} [/tex]

    This is my first attempt at anything like this so any help is welcome
     
  2. jcsd
  3. Sep 8, 2009 #2

    kuruman

    User Avatar
    Homework Helper
    Gold Member

    You are 100% correct. Congratulations on your first successful attempt. :approve:
     
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