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Homework Help: Qm problem

  1. Mar 13, 2008 #1
    1. The problem statement, all variables and given/known data

    I found the following problem in two places.But I doubt the first one is wrong.



    Let [tex]\ u_1(\ x )[/tex] and [tex]\ u_2(\ x )[/tex] are two degenerate eigenfunctions of the hamiltonian [tex]\ H =\frac{\ p^2 }{2\ m }\ + \ V (\ x )[/tex]



    Then prove that

    [tex]
    \int u_1(x)\left(x\frac{\hbar}{i}\frac{\partial}{\partial x}-\frac{\hbar}{i}\frac{\partial}{\partial x}x\right)u_2(x)dx=0.
    [/tex]

    and [tex]\ u_1 (\ x )[/tex] and [tex]\ u_2 (\ x )[/tex] are orthogonal to each other.

    OR prove that

    [tex]
    \int u_1*(x)\left(x\frac{\hbar}{i}\frac{\partial}{\parti al x}+\frac{\hbar}{i}\frac{\partial}{\partial x}x\right)u_2(x)dx=0.
    [/tex]



    2. Relevant equations



    3. The attempt at a solution

    Now I do not know what is correct expression.I found them in different places.May be both are correct.
    But in the first one the sandwitched operator (xp-px) is skew hermitian.Hence, its eigenvalues are imaginary!!!So I doubt about its validity.

    Now how to prove?As xp+px is hermitian, it is tempting to think that if [tex]\ u_1 (\ x )[/tex] and [tex]\ u_2 (\ x )[/tex]
    are non-degenerate eigenfunctions of this operator,then it is done.
    But I wonder, how to show it?
     
  2. jcsd
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