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Homework Help: Energy Eigenstate

  1. Jan 28, 2006 #1
    Trying to get my head around this problem and would very much appriciate any suggestions.

    Given a wavefunction \psi(X) i am asked if it is an energy eigenstate for a free particle moving in one dimension? Any suggestion on how I start a problem like this?


  2. jcsd
  3. Jan 28, 2006 #2
    Well, this is rather easy if you just know how to solve the Schrödinger equation in the case of a free particle (ie potential V(x) = 0)

  4. Jan 28, 2006 #3
    As was pointed out, a free particle means V(x) = 0.
    Furthermore, you'll be solving the one dimensional, time-independent S.E, since you're given psi(x).

    Most undergrad texts work this out at one point or another. I especially like Griffith's explanations - and it should help you a lot (it's done in position space, in 1D).

    In case you don't have it, to get you started:
    Write the SE:
    [tex]-\frac{\hbar ^2}{2m}\frac{d^2 \psi}{dx^2} = E\psi[/tex]
    Rearrange, then define
    [tex]k \equiv \frac{\sqrt{2mE}}{\hbar}[/tex]

    Being able to just "see" that you should define k as such, to make it easier (or possible?) to solve isn't something I was able to do. It would have taken me ages to find that on my own.

  5. Jan 30, 2006 #4


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    The spectral problem

    [tex] \hat{H}|\psi\rangle =E|\psi\rangle [/tex]

    in case of a free particle has a solution of the form

    [tex] \psi (x)=\langle x|\psi\rangle [/tex]

    , where [itex] \langle x| [/itex] is a tempered distribution and [itex] |\psi\rangle [/itex] is a test function.

    So you'll have to see whether your wavefunction can be obtained in this method: applying a linear functional on a vector from [itex] L^{2}\left(\mathbb{R}\right) [/itex].

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