Degenerate energy levels

Tiny things have energy.

Sometimes, different states have the same energy. This is called degeneracy. It means one energy level has many different ways to exist.

Think of a square box. A tiny particle can move in different ways inside it. Some of these ways use the exact same amount of energy.

If a system has symmetry, it often has these same energy levels. This makes it easier to study how things work. Finding these levels helps us understand the tiny world.

In the tiny world of quantum mechanics, things work differently than they do in our everyday lives. Everything has energy. Usually, if you know the energy of a tiny particle, you know exactly what state it is in. But sometimes, things are not so simple.

Sometimes, different states have the exact same amount of energy. This is called degeneracy. When this happens, knowing the energy is not enough to tell us what the particle is doing. We need other numbers to describe its state. The number of different states at one energy level is called the degree of degeneracy.

We can see this in a square box. A particle inside the box can move in different ways. Some ways use the same energy. For example, a particle might move more in one direction or the other. If the box is a perfect square, these two ways use the same power. This happens because the box has symmetry. Symmetry means the parts look the same in different ways. This symmetry often causes degeneracy.

In one dimension, like a single line, this rarely happens. But in two or three dimensions, it happens much more often.

In the tiny world of quantum mechanics, energy levels are very important. An energy level is a specific amount of energy a system can have. Usually, one energy value belongs to only one state. But sometimes, two or more different states have the exact same energy. Scientists call this situation degeneracy. When this happens, knowing the energy is not enough to describe the system. You need other special numbers to tell the states apart. The number of different states at one energy level is called the degree of degeneracy.

To understand how this works, we can look at a particle in a box. Imagine a particle moving inside a rectangular plane with walls it cannot pass. The energy of this particle depends on how it moves in different directions. We use numbers called quantum numbers to describe these movements. If the box is a perfect square, the energy levels become degenerate. For example, a particle could move more in the x-direction or the y-direction. In a square box, these two different paths use the same amount of energy.

Math helps us find these patterns in different dimensions. In a one-dimensional system, like a single line, degeneracy is very rare. For a particle moving in a one-dimensional potential, the degree of degeneracy never goes above two. It has been proven that there are no degenerate bound states in one dimension. However, things change when we move to two or three dimensions. In a three-dimensional cubic box, the energy depends on three different quantum numbers. Because you can swap these numbers, each energy level has a degeneracy of at least three.

We can see real examples of these patterns in science. Scientists study two-dimensional systems to understand much bigger, more complex things. Some real two-dimensional materials are made of very thin layers of atoms. Examples include systems made of Helium, Neon, Argon, or Xenon. These materials help us see how particles behave in a flat space. By using math models like the particle in a box, we can predict how many states will share the same energy. This helps us understand the building blocks of our world.

Why does this happen in the first place? The physical reason for degeneracy is often symmetry. Symmetry means the system looks the same even if you change it in certain ways. For example, a square looks the same if you turn it. This symmetry allows different states to share the same energy value. If a system has symmetry, we can often find the energy levels without doing hard math. Studying symmetry is a powerful way to learn how the universe is organized.

In the field of quantum mechanics, energy levels define the specific amounts of energy a system can possess. Usually, a single energy value corresponds to just one unique state. However, a phenomenon called degeneracy occurs when two or more different measurable states share the exact same energy value. The number of distinct states associated with a specific energy level is known as the degree of degeneracy. When degeneracy is present, measuring the energy alone is insufficient to identify the exact state of the system. Scientists must use additional quantum numbers to distinguish between these different states.

To understand this mathematically, we treat the possible states of a quantum system as abstract vectors. These vectors exist within a complex Hilbert space. Physical observables are represented by linear Hermitian operators that act upon these vectors. When an operator acts on a vector and results in the same vector multiplied by a scalar, that scalar is called an eigenvalue. The vector itself is known as the eigenvector. If an eigenvalue corresponds to two or more linearly independent eigenvectors, that eigenvalue is considered degenerate. The set of all these eigenvectors forms a subspace called an eigenspace. The dimension of this eigenspace tells us the degree of degeneracy.

Degeneracy behaves differently depending on the number of dimensions in the system. In a one-dimensional system, such as a particle moving along a single line, degeneracy is extremely limited. For a particle in a one-dimensional potential, the degree of degeneracy never exceeds two. In fact, it has been proven that there are no degenerate bound states for normalizable wave functions in one dimension. This is because the time-independent Schrödinger equation acts as an ordinary differential equation in this setting. This mathematical constraint prevents the complex patterns seen in higher dimensions.

As we move into two-dimensional systems, the complexity increases significantly. Scientists study two-dimensional quantum systems to understand more complex three-dimensional matter. Real-world examples include monoatomic layers on the surface of solids. Experimental examples include two-dimensional electron systems like MOSFETs or superlattices of Helium, Neon, Argon, and Xenon. A common model used to study these systems is the "particle in a box." In a rectangular two-dimensional box with impenetrable walls, energy depends on two quantum numbers. If the box is a perfect square, the dimensions are equal. In this square setup, certain pairs of states become degenerate because the energy values remain the same even if the quantum numbers are swapped.

In a three-dimensional cubic box, the energy levels depend on three different quantum numbers. Because these three numbers can be interchanged without changing the energy, every energy level has a degeneracy of at least three, provided the numbers are not all equal. This shows how the geometry of a space directly influences its quantum properties. For example, if the three quantum numbers are different, you can swap their positions in several ways to get the same energy. This creates a larger eigenspace and a higher degree of degeneracy. The more ways a system can be rearranged without changing its energy, the higher the degeneracy will be.

One way to manage this complexity is by finding a complete set of commuting observables. If an energy level is degenerate, simply knowing the energy does not provide a unique basis of eigenvectors. However, if we find another observable that commutes with the Hamiltonian, we can use it to label the states. Two operators commute if the order in which they are applied does not change the result. By choosing these additional commuting observables, scientists can construct an orthonormal basis. This allows them to assign unique labels to each state, even when they share the same energy level. This process is vital for characterizing the exact state of a quantum system.

The physical origin of degeneracy is often found in the symmetry of the system. Symmetry means the system remains unchanged under certain transformations, such as rotation or reflection. For example, the parity operator looks at whether a state is even or odd by changing the position vector to its negative. If a potential is even, the Hamiltonian is also an even operator. Studying these symmetries allows scientists to predict energy levels and degeneracy without solving the full Schrödinger equation. This connection between symmetry and degeneracy is a fundamental principle that helps simplify the study of the physical world.