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Degenerate Perturbation Theory (Particle in 3D box)

  1. Dec 3, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider a particle confined in a cubical box with the sides of length L each.
    Obtain the general solution to the eigenvalues and the corresponding eigenfunctions.

    Compute the degeneracy of the first excited state.

    A perturbation is applied having the form

    H' = V from 0 <x < ##\frac{L}{2}## and 0<y<##\frac{L}{2}##
    = 0 elsewhere

    Compute the first order shifts in the energy levels of the first excited state due to the perturbation.
    Construct the corresponding eigenstates.


    2. Relevant equations



    3. The attempt at a solution


    I'm struggling with perturbation theory.

    I solved the first part of the problem.
    the first excited state is triply degenerate
    and has Energy equal to:
    ## E = \frac{\pi^2\hbar^2}{2mL^2}(n_1^2+n_2^2+n3^2) = \frac{6\pi^2\hbar^2}{3mL^2}##


    ##\psi = (\frac{2}{L})^\frac{3}{2}sin(k_1x)sin(k_2y)sin(k_3z)##

    the quantum numbers have to start at n = 1

    so for the excited state the degeneracy is

    ##n_1 = 1, n_2 = 1, n_3 = 2 ## corresponds to ##\psi_1## and ##E_1##
    ##n_1 = 1, n_2 = 2, n_3 = 1 ## corresponds to ##\psi_2## and ##E_2##
    ##n_1 = 2, n_2 = 1, n_3 = 1 ## corresponds to ##\psi_3## and ##E_3##



    Now, I've used projection operators on the Hamiltonian and have gotten through a series of steps the following matrix:

    \begin{array}{cc}
    E_1 + <1|V|1> & <1|V|2> & <1|V|3> \\
    <2|V|1> & E_2 + <2|V|2> & <2|V|3>\\
    <3|V|1> & <3|V|2> & E_3+<3|V|3>
    \end{array}

    which is equal to the matrix:

    \begin{array}{cc}
    <1|i>\\
    <2|i>\\
    <3|i>
    \end{array}

    with ##E_i## multiplied in front (sorry, I don't know how to write that here)

    Now, I think I should bring that over and solve like a normal

    ## (A-\lambda I)x = 0## problem

    but what do I do with the subscript i?
    Do I drop it?

    I also don't know what to do from here.
    I tried asking my professor, but he just told me to "solve the integrals."

    What integrals??
    I'm so lost.

    Any help would really be appreciated.
    I know that matrix with the bras and kets should have lots of zeros, but I don't know how to get it like that...
     
  2. jcsd
  3. Dec 4, 2013 #2

    vela

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    How can a 3x3 matrix be equal to a 3x1 matrix? What did you get for the matrix elements?

     
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