# First-order correction to energy in perturbed harmonic oscillator [closed]

I know, from the perturbation theory, that, if I have the hamiltonian $$\hat H = \hat H_0 + \lambda \hat W$$ where $$\hat H_0$$ is the unperturbed hamiltonian of which I know its eigenvectors and eigenvalues, and $$W$$ is the perturbation. Then the energy of the perturbed hamiltonian, corrected to the first order, is given by $$E_n\approx\varepsilon_0+\lambda \varepsilon_1 \tag{1}$$ where $$\varepsilon_0$$ is the nth eigenvalue of $$H_0$$ (i.e. $$H_0|\varphi_n\rangle =\varepsilon_0 |\varphi_n\rangle$$) and $$\varepsilon _1=\langle \varphi_n|\hat W|\varphi_n\rangle$$.

My question comes from a specific problem in which the hamiltonian given to me is a peculiar "anharmonic" oscillator $$\hat H = \frac{p^2}{2m}+\frac{m\omega^2}{2}x^2+\alpha x + \beta p^2$$ In this case I don't have just one parameter $$\lambda$$ but two ($$\alpha$$ and $$\beta$$). What should the expressión (1) be here?

Thanks

PS:In addittion I would like to know how to solve this problem in exact way, or at least, if this is possible. I suppose it could work with a change of variables.

• Why not just go through a similar derivation that gave you what you cite at the beginning of your question? – Aaron Stevens Jun 12 '19 at 4:32
• You say something like proposing $E_n\approx \varepsilon_{0,0}+\alpha \varepsilon_{1,0}+\beta \varepsilon_{0,1}+\alpha \beta \varepsilon_{1,1}$ ?? – Ariel Jun 13 '19 at 20:21

$$\hat H = \left(\frac{1}{2m}+\beta\right)p^2+\frac{m\omega^2}{2}\left(x+\frac{\alpha}{m\omega^2}\right)^2-\frac{\alpha^2}{2 m \omega^2}$$
Define $$\bar{m}$$ to satisfy $$\frac{1}{2\bar m}=\frac{1}{2m}+\beta$$, define $$\bar\omega$$ to satisfy $$m\omega^2=\bar m\bar \omega^2$$, and define $$\bar{x}=x+\frac{\alpha}{m\omega^2}$$. Then the Hamiltonian becomes
$$\hat H = \frac{p^2}{2\bar m}+\frac{\bar m\bar\omega^2}{2}\bar{x}^2-\frac{\alpha^2}{2 m \omega^2}$$ which is just a SHO Hamiltonian plus a constant energy offset. So, you should be able to find the energy levels for this Hamiltonian, and write them out in terms of $$m,\omega, \alpha, \beta$$. Expanding the result to first order in $$\alpha,\beta$$ will immediately answer your first question.