Bound state negative potentials into harmonic oscillator basis

[GeneralGrant]

Hello readers,

Given the potential

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

I have found numerically 12 negative energy solutions

Now I want to try to solve for these using matrix mechanics

I know the matrix form of the harmonic oscillator operators X_ho, P_ho.

I believe I need to perform the task

<X | H | X> ??

to get a matrix form of H, then solve for its eigenvalues. But don't know how to use my H to get X and P operators or how to get my H into a matrix form with which to do the matrix algebra needed.


Looking around and in my 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 which has readily available matrix forms?

 

[DrDu]

You can express p and x in terms of a and $a^+$, the anihilation and generation operators in the HO basis.

 

[GeneralGrant]

I have done so for the harmonic oscillator, which is one way that i got my X and P for the HO. And have tried to use the X and P forms from the HO in my Hamiltonian in question, but because its 1/Sqrt(1+x^2) i always end up dividing by a zero for the none tri-diagonal terms.(using 1 = identity matrix)