1. The problem statement, all variables and given/known data 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 2. Relevant equations 3. 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.