# General wavefunction for a system of two coupled, quantum oscillators

Suppose we have two quantum harmonic oscillators, with different masses $$m_{1},m_{2}$$ and frequencies $$\omega_{1,2}$$. Then we can say particles are \emph{distinguishable}, in the sense that particle $$1$$ has a different mass ($$m_{1}$$) than particle $$2$$ ($$m_{2}$$); they are not identical. Then, if the two oscillators are not coupled to each other, we can write the wavefunction for a state $$|n,m\rangle$$ as: $$\begin{eqnarray} \Psi(x_{1},x_{2}) = H_{n}(x_{1})H_{m}(x_{2}) \end{eqnarray}$$ where $$H_{n}(x)$$ is the Hermite polynomial. Such wavefunction satisfies the time-independent Schrodinger equation with $$E = \hbar\omega_{1}\left(n+\frac{1}{2}\right) + \hbar\omega_{2}\left(m+\frac{1}{2}\right),$$ and represents a state where particle 1 is in the $$n$$th state of the oscillator, whereas particle $$2$$ would be in the $$m$$th state. I assume so far this is correct.

Now, consider the case where these two oscillators are coupled to each other, by some interacting term $$V(x_{1},x_{2})$$. Then, the global motion of the system is supposed to carry some entanglement, and the wavefunction does not have the simple product form as above. My question is on how could one proceed in order to find an exact representation of the wave-function in this case. Moreover, under which conditions is the system supposed to be integrable? meaning that one can compute the exact spectrum in the same way as one does for identical, indistinguishable particles (for instance in 1D, by means of Bethe ansatz methods or similars, although this is a 2-particle problem). Are there any known theorems on the integrability of such systems, for instance, considering anharmonic effects to infinite order of perturbation theory?

• I haven't actually worked it out, but I suspect you could get the wavefunction to be a product independent wavefunctions if you use a coordinate transformation to "decouple" them, very much like in the classical case, defining some $X = x_1 + x_2$ and $x=x_1-x_2$ (assuming both the oscillators are identical, with a coupling that only depends on their separation). – Philip Jun 14 at 17:19
• I think that might be true in a very specific case; for example when the two oscillators are identical (which is not the case in the example). In addition, one can have a complicated interaction potential V(x1,x2) depending on the coordinates, then in that case the rotation probably offers poor progress. Another thing I considered is a more general unitary transformation of the hamiltonian, that uncouples the V(x1,x2) part, but this is not easy I guess. – Zarathustra Jun 14 at 17:26
• Sutherland has a book called "Beautiful Models" which looks like it's about integrable potentials in one dimension. – Connor Behan Jun 14 at 18:18
• Thanks for the reference Connor, I know about Sutherlands book, mostly about Bethe ansatz, but it might be worthwhile taking a look to see if this problem is there! – Zarathustra Jun 14 at 19:55
• To get an idea of what systems are integrable or not, you might enjoy reading this Phys.SE post. – Qmechanic Jun 16 at 8:00