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, and 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 can be thought of as having 3 discrete energy levels. Humans are complex beings, they will find themselves on any of the floors without any particular reason. Electrons are inherently lazy and will exist in the lowest energy state, but due to temperature, electrons have some inherent energy and so depending on how different the energy levels are you might find a few electrons on the 2nd or even 3rd floor.

Ludwig Boltzmann was the individual that showed mathematically how the population of two states is dependent on the difference in the energy between the two states. The Boltzmann distribution is shown in eq. 2.2.

Activity

Reproduce the graph (left side) in Example problem 2.1 using Mathematica.