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Well, I know this have no sense. But I was trying to solve a problem on Cohen Tannoudji. The problem is in chapter IV, complement ##J_{IV}##, exercise 8. It says:

I transcribed the whole exercise, but I got stuck in part 1. I evaluated the matrix elements for W.

I've got the Hamiltonian:

##H= \begin{pmatrix}

E_0 & -a & 0 \\

-a & E_0 & -a \\

0 & -a & E_0 \\ \end{pmatrix} ##

When I diagnoalize it, I get the three eigenvalues: ##E_1=E_0,E_2=E_0-\sqrt{E_0^2+2a^2},E_3=E_0+\sqrt{E_0^2+2a^2}##

Then, I've tried to find the eigenstates,

For ##E_1##:

##\left | {E_1} \right >= \frac{1}{\sqrt{2}} \begin{pmatrix}

1 \\

0\\

-1\\ \end{pmatrix} ##

But then, when I try to get ##\left | E_1 \right >## I get intro trouble. I have the system of equations:

##(E_0-E)\beta-a\gamma=0 \\

-a\beta+(E_0-E)\gamma-a\eta=0 \\

-a\gamma+(E_0-E)\eta=0##

So I get ##\gamma=0,\beta=0,\eta=0##. I get the zero vector as the eigenvector. And that's absurd, because any eigenvalue could be an eigenvalue for the zero eigenvector. I don't know if I did something wrong, or if I have to interpret this result somehow.

Thanks in advance for your help. I have transcribed the whole exercise because, if I get some help and I can go on with it, I would like to discuss the other results that I could find.

Consider an electron of a linear triatomic molecule formed by three equidistant atoms. We use ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >## to denotate three orthonormal states of this electron, corresponding respectiveley to three wave functions localized about the nuclei of atoms A, B, C. We shall confine ourselves to the subspace of the state space spanned by ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >##.

When we neglect the possibility of the electron jumping from one nucleus to another, its energy is described by the Hamiltonian ##H_0## whose egienstates are the three states ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >## with the same eigenvalue ##E_0##. The coupling between the states ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >## is described by an additional Hamiltonian defined by:

##W\left | {\phi_A} \right >=-a \left | {\phi_B} \right > \\

W\left | {\phi_B} \right >=-a \left | {\phi_A} \right >-a \left | {\phi_C} \right > \\

W\left | {\phi_C} \right >=-a \left | {\phi_B} \right > ##

Where ##a## is a real positive constant.

1. Calculate the energies and stationary states of the Hamiltonian ##H=H_0+W##.

2. The electron at time t=0 is in the state ##\left | {\phi_A} \right >##. Discuss qualitatively the localization of the electron at subsequent times t. Are there any values of t for which it is perfectly localized about atom A,B, or C?

3. Let ##D## be the observable whose eigenstates are ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >## with respective eigenvalues ##-d,0,d##. ##D## is measured at time t; what values can be found, and with what probabilities?

4. When the initial state of the electron is arbitrary, what are the Bohr frequencies that can appear in the evolution of ##\left < D \right >##? Give a physical interpretation of ##D##. What are the frequencies of the electromagnetic waves that can be absorbed or emmitted by the molecule?Consider an electron of a linear triatomic molecule formed by three equidistant atoms. We use ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >## to denotate three orthonormal states of this electron, corresponding respectiveley to three wave functions localized about the nuclei of atoms A, B, C. We shall confine ourselves to the subspace of the state space spanned by ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >##.

When we neglect the possibility of the electron jumping from one nucleus to another, its energy is described by the Hamiltonian ##H_0## whose egienstates are the three states ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >## with the same eigenvalue ##E_0##. The coupling between the states ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >## is described by an additional Hamiltonian defined by:

##W\left | {\phi_A} \right >=-a \left | {\phi_B} \right > \\

W\left | {\phi_B} \right >=-a \left | {\phi_A} \right >-a \left | {\phi_C} \right > \\

W\left | {\phi_C} \right >=-a \left | {\phi_B} \right > ##

Where ##a## is a real positive constant.

1. Calculate the energies and stationary states of the Hamiltonian ##H=H_0+W##.

2. The electron at time t=0 is in the state ##\left | {\phi_A} \right >##. Discuss qualitatively the localization of the electron at subsequent times t. Are there any values of t for which it is perfectly localized about atom A,B, or C?

3. Let ##D## be the observable whose eigenstates are ##\left | {\phi_A} \right >, \left | {\phi_B} \right >, \left | {\phi_C} \right >## with respective eigenvalues ##-d,0,d##. ##D## is measured at time t; what values can be found, and with what probabilities?

4. When the initial state of the electron is arbitrary, what are the Bohr frequencies that can appear in the evolution of ##\left < D \right >##? Give a physical interpretation of ##D##. What are the frequencies of the electromagnetic waves that can be absorbed or emmitted by the molecule?

I transcribed the whole exercise, but I got stuck in part 1. I evaluated the matrix elements for W.

I've got the Hamiltonian:

##H= \begin{pmatrix}

E_0 & -a & 0 \\

-a & E_0 & -a \\

0 & -a & E_0 \\ \end{pmatrix} ##

When I diagnoalize it, I get the three eigenvalues: ##E_1=E_0,E_2=E_0-\sqrt{E_0^2+2a^2},E_3=E_0+\sqrt{E_0^2+2a^2}##

Then, I've tried to find the eigenstates,

For ##E_1##:

##\left | {E_1} \right >= \frac{1}{\sqrt{2}} \begin{pmatrix}

1 \\

0\\

-1\\ \end{pmatrix} ##

But then, when I try to get ##\left | E_1 \right >## I get intro trouble. I have the system of equations:

##(E_0-E)\beta-a\gamma=0 \\

-a\beta+(E_0-E)\gamma-a\eta=0 \\

-a\gamma+(E_0-E)\eta=0##

So I get ##\gamma=0,\beta=0,\eta=0##. I get the zero vector as the eigenvector. And that's absurd, because any eigenvalue could be an eigenvalue for the zero eigenvector. I don't know if I did something wrong, or if I have to interpret this result somehow.

Thanks in advance for your help. I have transcribed the whole exercise because, if I get some help and I can go on with it, I would like to discuss the other results that I could find.

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