Is the Given Interaction Term Correct for Coupled Harmonic Oscillators?

[VatanparvaR]

Homework Statement

Find the energy levels and the wave functions of two harmonic oscillators of mass $$m_1$$and $$m_2$$,
having identical frequencies $$\omega$$, and coupled by the interaction $$\frac{1}{2}k(x_{1}^{2}-x_{2}^{2})^2$$

Homework Equations

Schrodinger equation.

The Attempt at a Solution

My first question is, if the interaction term as it is given makes sense, as it is not of the characteristic of energy/ potential (unit is different!). Therefore i cannot see how to put it into the hamiltonian.
If i just take the interaction terms as it is and add it to the sum of two harmonic oscillators, the term gets long and i can not decouple this hamiltonian

Please give me any hint on this problems. Thank you very much in advance!

 

[VatanparvaR]

No one can help ? :)

Now I got that from Prof. that it was given wrong, it must bu (x1-x2)^2 in the interaction term, but if it is like above given interaction term, is it right? I mean is such an interaction term possible?