Bound States, Negative Potential, Alternate Basis, Matrix Mechanics

[GeneralGrant]

Homework Statement

Given the potential

V(x) = - 1/ sqrt(1+x^2)

Consider this in a 50x50 matrix representation of the hamiltonian in the basis of a one dimensional harmonic oscillator. Determine the eigenvalues and eigenvecotrs, the optimal parameter for the basis, and cop ate the results to the previous problem. plot the eigenfunctions and check the orthogonality.

Hbar = mass = 1

Homework Equations

The Attempt at a Solution

the previous problem is to use a shooting method to find the first 10 eigenstates which i have done, and plotted the Eigenfunctions. E = -.669, -.274, -.157, -.09 . . .

But this part actually uses quantum mechanics which i don't know very well.

I believe I need to perform the task

<O | H | O> ??

such that O is an operator in the Harmonic oscillator basis. But i don't know how to do that since the operators are matrices, and my Hamiltonian is a differential equation. I know the matrix form of the X and P operators of the harmonic oscillator, and i have done this for the problem being that of an alternate harmonic oscillator, e.g., describe H(w=.5) in basis of H(w=1). in that problem i just replaced the X and P operators with the ones form the other Basis.

But when i tried to do this for the hamiltonian

H = P^2/2m -1 /sqrt(1+X^2) with X and P the Harmonic oscillator operators, there is a division by zero that ruins the results for all the zero terms in X^2.


Looking around and in my old books, I'm fining Matrix information and Equational information but not so much on transporting between the two(except for the HO of course which is every where and partially why i want to use it)

How does one perform perform these Bra Ket actions?
does one transform a hamiltonian into its own operator form first or can i use the operators from a Harmonic Oscillator?
Is there a way to represent this hamiltonian in the HO basis using the matrix forms of the HO?


Note this problem is not from a Quantum Physics course, but from a computational course.

 

[DrClaude]

What you have is a Hamiltonian $\hat{H}$, of which you need to find the eigenvalues $E_n$ and eigenvectors $\psi_n$,
$$
\hat{H} \psi_n = E_n \psi_n
$$
The wave function is to be written in terms of a basis of harmonic oscillator functions
$$
\psi_n = \sum_i c_{n,i} \phi_i
$$
where the $c_{n,i}$ are (complex) coefficients and $\phi_i$ are eigenfunctions of the harmonic oscillator Hamiltonian $\hat{H}_{\mathrm{ho}}$
$$
\hat{H}_{\mathrm{ho}} \phi_i = E_i \phi_i
$$
In this basis, your original Hamiltonian $\hat{H}$ can be expressed as a matrix $H$, with elements
$$
H_{ij} = \langle \phi_i | \hat{H} | \phi_j \rangle
$$
This is what you need to calculate. I'm not sure how you are supposed to go about it. The $\phi_i$ are easy to find in any QM textbook, but I'm not sure if you can get analytical expressions for the $H_{ij}$ or if you have to calculate them numerically.

Once you have $H$ in matrix form, the eigenvectors you find are going to be vectors of coefficients $c_n$.

 

[GeneralGrant]

I have to solve it numerically, I'm pretty sure of that.
But i know know what the expression
$$
< O | \hat{H} | O >
$$
means other then it is an inner product of the matrix $\hat{H}$ and O and then a product with the transpose of O.

I think its supposed to representable as an integral somehow? But i doubt that is what we are supposed to do.

Previously in the course: we did the following

X(in the w=.5 basis) = some matrix a
p(in the w=.5 basis) = some matrix b

Then

H(in the basis w=1) = (1/2m)(matrix multiplication of(b with b(transpose)) + m w^2 /2 (matrix multiplication of (a with a(transpose))

which yielded a none diagonal matrix, if the basis were the same it would be diagonal, and then we solve it for the diagonal matrix and get the awnser.

Can i use a and b in my new H similar to how we did before, they said that this old problem was a hint at doing this one?