Two simple harmonic oscillators, each of natural frequency ω, are coupled in such a way that there is no interaction between them if the oscillators have different quantum numbers, whereas their combined energy is (2n+1)ℏω+Δ if the oscillators have the same quantum number n. The system is in thermal equilibrium at temperature T. Find the probability that the two oscillators have identical quantum numbers. Find and interpret the zero-temperature limit of your result, for all values of Δ.