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The spectrum above is an approximation to the P and R branches of the rotationalvibrational absorption spectrum of CO. This is a spectrum for the case of a transition from the vibrational ground state to the first excited vibrational state. Each vibrational energy level, however, contains a number of rotational energy levels because the energy between rotational energy levels is in the 10^{−4} eV range while the energy difference between vibrational levels is 10^{−1} eV. So, as the molecule goes from ω_{0} to ω_{1} (ground to first excited vibrational states), there is also a transition between rotational states (governed by the selection rule ΔJ = +/−1 where J is the angular momentum quantum number for the molecule. The P branch are transitions where ΔJ = −1 and the R branch, ΔJ = +1. The intensity depends on the initial population of the states. The rotational energy of the molecule is given classically by E = L^{2}/2I (kinetic energy of rotation where I is the moment of inertia of the molecule and L is the classical angular momentum). The rotational levels are quantized and we will use J as the angular momentum quantum number so that the energy of rotation is given by ε_{J }= J(J + 1)ħ^{2}/2I.
Problem based on Example 9.2, Beiser, Modern Physics 6th ed., McGraw Hill, 2003, p. 300.