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Homework Help: Triatomic molecule

  1. Nov 11, 2007 #1
    1. The problem statement, all variables and given/known data
    Consider an electron of a linear triatomic molecule formed by three equidistant atoms. We use [tex] |\phi_A>, |\phi_B>, |\phi_C>[/tex] to denote three orthonormal staes of this electron, corresponding respectively to three wave functions localized about the nuclei of atoms A, B and C. We shall confine ourselves to the subspace of the state space spanned by [tex] |\phi_A>, |\phi_B>, |\phi_C>[/tex].

    When we neglect the possibility of the electron jumping from one nucleus to another, its energy is described by the Hamiltonian [tex]H_0[/tex] whose eigenstates are the three states [tex] |\phi_A>, |\phi_B>, |\phi_C>[/tex] with the same eigenvalue [tex]E_0[/tex]. The coupling between states [tex] |\phi_A>, |\phi_B>, |\phi_C>[/tex] is described by an additional Hamiltonian W defined by:

    [tex] W|\phi_A> = -a|\phi_B>[/tex]
    [tex] W|\phi_B> = -a|\phi_A> - a|\phi_C>[/tex]
    [tex] W|\phi_C> = -a|\phi_B>[/tex]

    where a is a real positive constant.

    a) Calculate the energies and stationary states of the Hamiltonian [tex] H = H_0 + W[/tex].

    2. The attempt at a solution

    [tex] H_0 = E_0 \[ \left( \begin{array}{ccc}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & 1 \end{array} \right)\] [/tex]

    [tex] W = -a\[ \left( \begin{array}{ccc}
    0 & 1 & 0 \\
    1 & 0 & 1 \\
    0 & 1 & 0 \end{array} \right)\] [/tex]

    [tex] H = H_0 + W = E_0 \[ \left( \begin{array}{ccc}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & 1 \end{array} \right)\] [/tex] [tex]-a\[ \left( \begin{array}{ccc}
    0 & 1 & 0 \\
    1 & 0 & 1 \\
    0 & 1 & 0 \end{array} \right)\] [/tex]

    [tex] H = \[ \left( \begin{array}{ccc}
    E_0 & -a & 0 \\
    -a & E_0 & -a \\
    0 & -a & E_0 \end{array} \right)\] [/tex]

    When I try to calculate the eigenvalues of this matrix to get the energies, I end up with an algebraic mess that involves a cubic function for [tex]\lambda[/tex] and I'm not sure how to solve, so I think I'm probably on the wrong track.
     
  2. jcsd
  3. Nov 11, 2007 #2

    Dr Transport

    User Avatar
    Science Advisor
    Gold Member

    You're on the right track, you have to solve the cubic function in [tex] \lambda [/tex] to get the eigenenergies and their associated wave functions.
     
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