More than one distinct quantum state of a system can be on the same energy level because energy is not the only measurable quantity that can be used to describe a state. For example, an electron orbiting a hydrogen atom has 8 states with energy -3.4 eV because of intrinsic spin, and angular momentum projections that do not affect the energy of the state. Here we consider the effect of such degeneracies (in a molecule) on the probability to find the molecule in a particular state.

The molecule can be in one of six states. It exchanges energy with a thermal reservoir of other molecules at a temperature T. There is one state with energy E₁ = k_BT, two degenerate states with energy E₂ = 3k_BT, and three degenerate states with energy E₃ = 7k_BT. [We have chosen these energies to simplify the calculations. In reality, the energy levels depend on quantum mechanics, not on the temperature.]

What is the probability that the molecule has energy E₂?
P(E₂) =