# Modified quantum harmonic oscillator spectrum and eigenstates

I am trying to find the eigenstates/eigenvalues of the following Hamiltonian $$\hat{H} = \hbar \omega \Big(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\Big)+A\big(\hat{a}^{\dagger}\hat{a}^{\dagger}+\hat{a}\hat{a}\big)\equiv \hat{H}_1+\hat{H}_2,$$ where $$\hat{a}$$ and $$\hat{a}^{\dagger}$$ are the standard ladder operators, with $$[\hat{a}, \hat{a}^{\dagger}] = 1$$.

My attempt

(1) I tried to check whether the two terms were commuting, to see if the problem admitted a separable solution, but $$\big[\hat{H}_1, \hat{H}_2\big]= \hat{a}^{\dagger}\hat{a}^{\dagger}-\hat{a}\hat{a}$$.

(2) I tried the trial solution (even states) $$|\phi\rangle = \sum_n c_{n2}|2n\rangle$$ which when substituted gives $$\hat{H}|\phi\rangle = \sum_n \left[\frac{\hbar \omega}{2}c_{2n}+\hbar \omega (2n) c_{2n}+c_{2n-2}+c_{2n+2}\right]|2n\rangle$$ which gives the infinite system $$c_{2n}= \left(\frac{\hbar \omega}{2}+2n\hbar \omega\right)c_{2n} +c_{2n-2}+c_{2n+2}.$$ I am not sure if this leads anywhere.

Question

Is there a standard way to approach this problem? Any help?

• This is a standard kind of problem in perturbation theory, where it is straightforward to get the second-order corrections to the energy. Getting the exact solution is much trickier.
– Buzz
Jun 7, 2022 at 12:24
• $A$ is not small though, so in general I can't use perturbation theory I think. Jun 7, 2022 at 12:31

The operators $$\hat C=\hat a^\dagger \hat a + \hat a\hat a^\dagger$$, $$\hat A=\hat a^\dagger \hat a^\dagger$$ and $$\hat B=\hat a\hat a$$ span an $$\mathfrak{su}(1,1)$$ Lie algebra.
The problem is solved using an $$SU(1,1)$$ transformation $$T$$ (in your case that's just a Bogoliubov transformation) that will map $$\hat A$$ to a linear combination of $$\hat A,\hat B$$ and $$\hat C$$. The same transformation will also map $$\hat B$$ and $$\hat C$$ to linear combinations of $$\hat A,\hat B$$ and $$\hat C$$. One then chooses the parameters in $$T$$ to eliminate the non-diagonal terms.
There's another equivalent approach which is to write your Hamiltonian in terms of $$x$$ and $$p$$. With some elementary insight you should see how this is actually the Hamiltonian of a harmonic oscillator with frequency $$\omega^\prime$$ which is a multiple of the original frequency $$\omega$$ of your problem. Once $$\hat H$$ is expressed in terms of $$\omega^\prime$$ you can just quantize that in terms of new raising operators $$\hat b$$ and $$\hat b^\dagger$$.
It shouldn't be too hard for you to find literature on Bogoliubov transformations, or $$\mathfrak{su}(1,1)$$ (or equivalently $$\mathfrak{sp}(2,\mathbb{R})$$).
In fact, the Wikipedia article on Bogoliubov transformations is not a bad place to start. You should be able to invert $$\hat b=u\hat a+v\hat a^\dagger\, ,\qquad \hat b^\dagger=u^*\hat a^\dagger+v^*\hat a$$ and re-express your $$H$$ in terms of $$\hat b$$ and $$\hat b^\dagger$$, then choose $$u$$ and $$v$$ to eliminate the $$\hat b\hat b$$ and $$\hat b^\dagger\hat b^\dagger$$.