Discrete Energy levels and Boltzmann Distribution

Many classical physics problems calculate the energy of an object. The potential energy of an object, like a baseball, is proportional to the height of the ball relative to a surface, like the ground. Since the ball can exist at any height, the potential energy (PE) can have any value between PE-max and 0 (on the ground). Likewise the kinetic energy (KE) of a baseball is proportional to the speed or velocity of the ball and can have a value between KE-max and 0 (at rest). The PE and the KE represent a continuum of energy states.

Like a baseball, an electron also has a continuum of PE and KE when it is not confined. When there exist boundaries, like those within an atomic structure, the energy of the electron becomes restricted and we describe such a system as having discrete energy states. This existence of discrete energy states is a unique property of quantum mechanics.

For a moment think about a hypothetical situation involving CSB. CSB has 3 discrete levels. If we measured the potential energy of any given individual in the CSB it would be one of three values, PE1, PE2, PE3, where PE1 represents the potential energy while on floor 1, as so on. Although there are people transitioning from floor 1 to 2 and 2 to 3 during a passing period, very seldom, if ever, do we find an individual doing there school work on the stairs (or between the energy levels). CSB has 3 discrete energy levels. Humans are complex being, if there were no classes on the 3rd floor there would be no reason for anyone to go to the 3rd floor. Electrons are inherently lazy and will exist in the lowest energy state unless

If humans behaved as electrons do, they might all exist on the first floor, ie. the lowest energy state, since there is no apparent reason for them to be on the second floor.