1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?



Similar Discussions: Qm problem
  1. Problem is. (Replies: 0)

  2. MCQ problem (Replies: 0)

  3. Scattering problem (Replies: 0)

Loading...