Modification to the simple harmonic oscillator

[jamesonWHIS]

Homework Statement

The simple harmonic oscillator Hamiltonian is altered such that the p' = p + 2mcx. How does this affect the condition necessary for the matrix elements <m|x|n> and <m, x^2| n> to be nonzero, given |n> is an eigenstate of the original harmonic oscillator.

Relevant Equations

x = Sqrt(h/2mw)(a + adagger)

I was assuming there could be something via perturbation theory? I am unsure.

 

[DrClaude]

At first glance, I don't think that perturbation theory is necessary. The $\ket{n}$ form a complete basis, even for the modified Hamiltonian.

However, I do not understand the question. "Given $\ket{n}$ is an eigenstate of the original harmonic oscillator," then $\braket{m|\hat{x}|n}$ and $\braket{m|\hat{x}^2|n}$ are unchanged, whatever the Hamiltonian is.

 

[songanh]

At first glance, I don't think that perturbation theory is necessary. The $\ket{n}$ form a complete basis, even for the modified Hamiltonian.

I would like to question this statement. How do you know such modification doesn't change the size of the Hilbert's space?