Potential-energy function of diatomic molecule

[gregje]

Hi there all, I have this problem which I have issues with; there's some stuff I need to do in C and any help would be much appreciated.

For V(o) = 36 i need to find the ground state energy and normalised ground state function using matrix methods. I am allowed to use Matlab to find the eigenvalues and vectors.
The matrix method includes numerical techniques where there's finite approximations.

This picture is a general solution; for this specific problem the potential V(i) = V(x(i)) and lambda is equal to E.
Through finite approximations using Taylors rule you get the matrix

Im guessing that the ground state energy is the eigenvalue for when phi(0) = 0 and the other eigenvalue will be when phi(L) = 0. I am guessing that the normalised ground state function would be the eigenvector of this matrix?
So through the theory of eigenvalue and eigenvectors, deltaxsquared*lambda will be an eigenvalue and the matrix phi(1) phi(2) etc is an eigenvector.
I effectively need to calculate the eigenvalues and eigenvectors of a symmetrix tridiagonal matrix... basically a Hermitian matrix and I am aware that the process for a Hermitian matrix is a lot simpler than for anti symmetric. NAG routines however are unfamiliar to me (they are meant to be used) I am allowed to use Matlab to calculate the eigenvalues and eigenvectors.
I also need to write a C program to find the ground state energy and normalised ground state function using the matching method. I am completely unfamiliar with the matching method in C and I am not being given a lot of help. Apparently you have to start with 2 independant solutions, rescale one curve so that they cross and vary E until both curves have the same slope at the crossing point. I am aware however that this is very ambiguous so any help would be very appreciated.

 

[Jhero]

Hello there,

Thank you for reaching out for help with your problem. It sounds like you are trying to find the ground state energy and normalised ground state function for a potential V(o) = 36 using matrix methods.

To start, I would recommend using Matlab to find the eigenvalues and eigenvectors of the matrix. This will give you the ground state energy and normalised ground state function.

To do this, you will first need to construct the matrix using the potential V(x) and the finite approximations using Taylor's rule. Then, use the built-in function "eig" in Matlab to find the eigenvalues and eigenvectors.

Once you have the eigenvalues and eigenvectors, you can determine which one corresponds to the ground state energy and normalised ground state function. The ground state energy will be the eigenvalue for when phi(0) = 0 and the other eigenvalue will be when phi(L) = 0. The normalised ground state function will be the eigenvector for the ground state energy.

It is important to note that the matrix you have constructed should be Hermitian, so you can use the simpler process for finding eigenvalues and eigenvectors.

If you are not familiar with using NAG routines, it is perfectly fine to use Matlab for this step. However, if you would like to learn more about NAG routines, there are many resources available online that can help you get started.

As for writing a C program to find the ground state energy and normalised ground state function using the matching method, it may be helpful to break down the steps and understand each one before trying to write the program. The matching method involves finding two independent solutions and then varying the energy until both solutions have the same slope at the crossing point. It may also be helpful to consult with a colleague or mentor who is familiar with the matching method in C.

I hope this helps you in your research. Best of luck with your project!