If we look at something like electrons, we can see that we can only have certain energy levels for the electrons, n-1, n=2, n=3, ... As it turns out, the same is true for translational, rotational, and vibrational energy. These can only exist in distinct and discreet states. Let's take some time to look at these levels in each of the three forms and see what it means for the molecule as a whole.
Beginning with translational energy, we can see that the energy states are so close as to be almost indistinguishable. Let's take a bead and put it on a wire connecting two points. The bead is allowed to slide in wave function along the wire. Obviously, like any particle, it will move in wave motion, and energy states of the predicted nature will be observed (see link below eventually). If a is the distance between the two points, then the number of antinodes (n) can be used to form an interesting formula:
n * lamda / 2 = a (with n never equal to zero)
and now remember that lambda, being a wavelength, is equal to h/(m * v), where h, Planck's constant, is 6.63 * 10^-34 J * sec. Therefore, we can rewrite the above equation as
a = n/2 * h/(mv) and eventually as
1/2 mv^2 = (n^2 * h^2)/(8ma)
if you recall, one-half m v squared is equal to kinetic energy, so we can see that yes, translational energy does have different distinct and discreet energy levels. But look at them. If m and a are large, it is a very small amount of energy separating two levels. Overall, it is still a small space between levels owing to the 10^-34 of Planck's constant.
Now let's move onto rotational energy. Instead of putting a bead on a string between two points, put it on a rope that connects to itself in a circle. For the loop to match up, or have an initial zero at the same location, n lambda must equal two pi r. Note that in this circumstance n can equal zero. Eventually, using the same logic and manipulations from above, we can derive:
K = (n^2 * h^2) / (8 * pi^2 * mr^2) ,which yields an energy difference around 10^-23 Joules.
will be finished later because I'm taking a break