1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Hermitian Operators and Inner Products

  1. Oct 18, 2009 #1
    1. The problem statement, all variables and given/known data

    Consider the vector space of square-integrable functions [tex]\psi(x,y,z)[/tex] of (real space) position {x,y,z} where [tex]\psi[/tex] vanishes at infinity in all directions. Define the inner product for this space to be

    [tex]<\phi|\psi> = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \phi^* \psi dx dy dz [/tex]

    Show that the operator

    [tex]\hat{L_z} = -i \hbar \left(x \frac{\delta}{\delta y} - y \frac{\delta}{\delta x}\right)[/tex]

    is a self-adjoint (Hermitian) operator on this space.
    2. Relevant equations

    3. The attempt at a solution

    So I know that for an operator to be Hermitian that [tex]\hat{H} = \hat{H}^\dagger[/tex]

    so just starting from what I know...

    [tex]<\hat{L} \phi|\psi> = <\phi|\hat{L}^\dagger|\psi>[/tex]

    [tex]<\hat{L} \phi|\psi> = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \left(-i \hbar \left(x \frac{\delta}{\delta y} - y \frac{\delta}{\delta x}\right) \phi \right)^* \psi dx dy dz = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} i \hbar \left(x \frac{\delta}{\delta y} - y \frac{\delta}{\delta x}\right) \phi^* \psi dx dy dz[/tex]


    [tex]<\phi|\hat{L}^\dagger|\psi> = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \phi^* \left(i \hbar \left(x \frac{\delta}{\delta y} - y \frac{\delta}{\delta x}\right)\psi \right) dx dy dz = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} i \hbar \left(x \frac{\delta}{\delta y} - y \frac{\delta}{\delta x}\right) \phi^* \psi dx dy dz[/tex]

    But I guess these are my two main questions:

    For a non-matrix operator

    [tex]\hat{L_z} = -i \hbar \left(x \frac{\delta}{\delta y} - y \frac{\delta}{\delta x}\right)[/tex]


    [tex]\hat{L_z}^\dagger = i \hbar \left(x \frac{\delta}{\delta y} - y \frac{\delta}{\delta x}\right)[/tex]


    and, can I move around my partials in the second equation to match that of my first or do I have to carry them out?

    Thanks, and I'm sorry if what I said is complete junk seeing as how I'm not THAT strong in linear algebra.... yet.
  2. jcsd
  3. Oct 19, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    Actually, [itex]L_z^\dagger = L_z[/itex] -- that is what you are trying to show (it's precisely what being Hermitian means).

    So the idea is that you start from
    [tex]\langle L_z \phi \mid \psi \rangle[/tex]
    where L works on the first function, and use partial integration to show that it is equal to
    [tex]\langle \phi \mid L_z \psi \rangle[/tex]
    where L works on the second function.

    (Then, of course, since you know that it is also equal to [itex]\langle \phi \mid L_z^\dagger \psi \rangle[/itex] you can conclude that L is equal to its Hermitian conjugate and you are done).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook