In this blog post, we’ll use the familiar example of stairs to help you easily understand the quantization of energy—a core concept in quantum chemistry.
Recently, as companies like Google have been advancing quantum computing technology, the concept of “quantum”—and by extension, quantum chemistry—has once again garnered significant attention. However, quantum chemistry is also a field that many students find difficult to grasp. I, too, experienced a great deal of confusion when I first encountered quantum chemistry. Even among students interested in chemistry, it is not uncommon for them to struggle with quantum chemistry and lose interest as they study it. For this reason, in this post, I aim to explain the most fundamental concept of quantum chemistry—the discontinuity of energy—using the familiar example of stairs.
The most fundamental concept and foundational principle of quantum chemistry is the discontinuity of energy. Many people find it difficult to grasp the idea that energy is not continuous. In particular, for those who do not major in chemistry or physics, the concept of energy being discontinuous can feel somewhat unfamiliar and challenging.
This discontinuity of energy can be easily understood by thinking of a staircase. Let’s imagine a long staircase. Suppose there is a ball resting on one of the steps. For the ball to move from one step to the next, it must move a distance equal to the height of a single step. Conversely, the ball cannot remain in the space between steps. Energy works in the same way. While the height of the steps or the starting position may vary depending on the type and state of the atom or molecule, the fundamental property—that a ball cannot exist between steps—remains unchanged. Thus, atoms cannot have continuous energy states; they can only have discrete energy states. This is precisely what is meant by the quantization of energy.
Now, let’s look at this staircase from a great distance. If we’re so far away that we can’t see the individual steps, the staircase will look like a single gentle slope. Furthermore, the process of rolling the ball up the stairs will appear as if the ball can freely occupy any position along that slope. This is why we do not easily perceive the quantization of energy in our daily lives. In fact, because the quantization intervals of energy at the atomic level are very small, we—living in the macroscopic world—perceive energy as changing continuously.
However, there is an important difference between the staircase and the slope. If you place a ball on the slope, it rolls down to the lowest point. On the other hand, on a staircase, the ball does not roll all the way down to the floor but stops on the lowest step. Therefore, the ball on the staircase is one step higher than the ball at the lowest point of the slope. This is one of the fundamental concepts in quantum chemistry that explains various phenomena.
So how did we come to realize that this slope is actually made up of very small steps? At first glance, it is not easy to distinguish between the two structures. So how did scientists determine that this structure is actually like a staircase? Representative experiments supporting the quantization of energy include the blackbody radiation experiment, measurements of the specific heat of matter at low temperatures, and observations of atomic line spectra. Among these, the easiest example to understand is the line spectrum of the hydrogen atom.
Let’s return to the staircase example. Suppose there is a way to precisely measure the heights at which a ball can exist. If the structure were an incline, the ball could exist at any height. However, if the structure were a staircase, the ball could exist only at specific heights and not at any heights in between. Even if the steps were very low, the gaps between possible heights would narrow, but the fact that the heights are discrete would remain unchanged. Scientists have discovered that energy is quantized based on this principle. In fact, when measuring the energy spectrum of a hydrogen atom, a discrete line spectrum is observed—one that appears only at specific values rather than in a continuous form. Through this, scientists learned that the energy states of a hydrogen atom are discrete and that this property applies universally to other atoms as well.
This fundamental concept is the very starting point of quantum chemistry. The Schrödinger equation and the concept of orbitals—both core theories of quantum chemistry—were also developed based on this fundamental principle of energy quantization. Therefore, understanding this principle is of great help in grasping the core concepts of quantum chemistry. The simple structure of a staircase, which we encounter frequently in daily life, can serve as an excellent analogy for explaining an important principle of quantum chemistry. I hope this will help more people understand the basic concepts of quantum chemistry and, to some extent, dispel the preconception that quantum chemistry is merely a difficult field of study.
