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!

Another 2 questions in perturbation theory.

  1. Nov 28, 2009 #1

    MathematicalPhysicist

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    1. A particle of mass M is in a square well, subject to the potential:
    [tex]V(x)= V0\theta(x-a/2)[/tex] for x in (0,a) and diverges elsewhere, where theta is heaviside step function.
    In perturbation theory, find O(V0^2) correction to the energy and O(V0)to the eigenstate.

    2. A particle is moving in a two dimensional harmonic oscillator with perturbation [tex]\delta V=\lambda x^2 y^2[/tex] For the ground state and the first excited state, find the first order correction to.


    2. Relevant equations



    3. The attempt at a solution
    1. As I see it the hamiltonian has a perturbation in x in [a/2,a), which is V0.
    , but from the equation for the second order correction to the energy which is:
    [tex]\sum_{m doesn't equal n} \frac{|<m|V0|n>|^2}{E_n-E_m}[/tex], where [tex]E_k=\frac{(\hbar k)^2}{2M}+V_0[/tex], so that means the second order correction is zero cause <m|n> for m not equal n equlas zero, doesn't make sense to me, what am I missing?

    For 2, for the first excited state I get that it's degenrate, cause [tex]E_{1,0}=E_{0,1}[/tex], not sure how to find for the first excited state its first order correction.

    P.S
    I sent an email to the TA with my questions, so far he hasn't replied, this is why I am asking here.
     
  2. jcsd
  3. Nov 28, 2009 #2

    diazona

    User Avatar
    Homework Helper

    For #1, you can't just assume that [itex]\left\vert \langle m \vert V_0 \vert n\rangle\right\vert = 0[/itex] whenever [itex]m \neq n[/itex]. Remember how you compute a matrix element:
    [tex]\langle m \vert V_0 \vert n\rangle = \int\psi^{*}_m(x) V_0(x) \psi_n(x)\,\mathrm{d}x[/tex]
     
  4. Nov 29, 2009 #3

    MathematicalPhysicist

    User Avatar
    Gold Member

    Ok thanks.
    Can someone help me with question 2?
     
  5. Dec 1, 2009 #4

    MathematicalPhysicist

    User Avatar
    Gold Member

    bump, question 2?
     
  6. Dec 1, 2009 #5

    MathematicalPhysicist

    User Avatar
    Gold Member

    I still get that it's zero.
    What are eigenfunctions of this problem?

    I think that they are:
    [tex]\psi_n(x)=\sqrt(\frac{2}{a})sin(nx\pi/a)[/tex] but if it's so then <n|V|m>=0.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Another 2 questions in perturbation theory.
Loading...