Constructing Eigenstate for a given hamiltonian

[mahondi]

Homework Statement

Hi; I am trying to construct eigenstate for the given hamiltonian. I have the energy eigenvalues and corresponding eigenvectors. But How can I construct eigenstates?

Homework Equations

The Attempt at a Solution

I tried to use the H . Psi= E . Psi eigenvalue relation where Psi is the state that I want to construct.Namely, what is the procedure that I should follow. I'm glad if you answer to the question.

 

[Tangent87]

What is the Hamiltonian you are considering? I can't see it.

 

[LeonhardEuler]

When you solve [itex]H\psi=E\psi[/tex] you will get eigenvalues and what are usually called eigenfunctions or eigenstates. Eigenvectors usually refer to the vectors obtained from the ordinary algebraic eigenvalue problem with a matrix. (i.e. with a matrix corresponding to a system of linear algebraic equations, not differential equations). Maybe you already have the eigenstates, but are calling them the wrong word?

 

[mahondi]

I attached my detailed problem as a pdf file. Since this is my research project, I constructed Hamiltonian and found its eigenvalues and eigenvectors.In other words, I haven't solved the H . Psi=E . Psi equation yet.And I thought I have to solve Time dependent Schrödinger equation because Hamiltonian in the attachments depend implicitly on time. Now I have to find eigenstates of the hamiltonian to calculate the berry phase.

Sorry for the foggy question.

 

[fzero]

That Hamiltonian is not Hermitian, so you made some sort of mistake in converting the operator to matrix form. If you can't find your mistake, post some more details and maybe we can help.

You have solved the $$H\psi=E\psi$$ equation by finding the eigenvalues and eigenvectors. You do have to use the energy eigenvalues for the states to solve the time-dependent Schrodinger equation

$$i\hbar \frac{\partial \psi(t)}{\partial t} = E \psi(t)$$

for the time-dependence of the states. This should give you an equation to solve for $$\varphi$$. Again, give it a try and come back with questions.

 

[mahondi]

Hi, Thank you all.I made a mistake in calculations now my hamiltonian is hermitian and I realized that I have already found the eigenstates by finding eigenvectors of my hamiltonian:)but there is another question that I would like to ask. I want to simplify the expression given in the attachments. I tried to simplify it in mathematica by using TrigFactor code. My question is that Can I make more simplifications to this expression?

Thank you..