(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Triatomic molecule

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