What is a Bose-Einstein Condensate?

A Bose-Einstein Condensate (BEC) is considered the fifth state of matter. Bose-Einstein Condensation takes place when bosons are cooled to nearly absolute zero. When they are cooled to this point, all of the particles begin to occupy the same energy state and are indistinguishable from each other. This state of matter pretty much creates a super-atom. BECs have been successfully created in labs; however, BECs were hypothesized long before the first physical BEC in the mid-90s. We can use quantum mechanics to predict BECs.

What is a particle?

Matter is made up of particles, but what is a particle? In physics, atoms, protons, and electrons are often referred to as particles; however, this classification can become quite confusing. When we talk about particles we must differentiate between elementary and non-elementary particles. Elementary particles are the smallest known building blocks of the universe. They are thought to have no internal structure, meaning that researchers think about them as zero-dimensional points that take up no space (1). Neither atoms nor protons are considered to be fundamental despite their categorization as particles. Atoms are not fundamental because they are made up of protons, neutrons, and electrons. Protons are not fundamental because they are made up of smaller, fundamental particles called quarks and gluons. There are four major categories of elementary particles: quarks, leptons, gauge bosons, and scalar bosons. Quantum mechanics predicts that these four categories can be grouped into two larger categories: fermions and bosons.

Modeling an Atom

When we picture an atom, we often imagine “Figure 1,” electrons orbiting a nucleus of protons and neutrons much like planets orbiting a star; however, this model is misleading. According to the Heisenberg Uncertainty Principle, there is a limit to what we can know about a quantum system. For example, the more precisely we know the position of a particle, the less we know about the other properties of that particle such its momentum. “Figure 2” facilitates a better understanding of an atom from a quantum mechanical perspective. In “Figure 2,” the blue dots represent electron probabilities. The darker areas represent the positions in which the electrons are more likely to be located.

Electron Configuration

We all remember electron configuration: 1s, 2s, 2p, 3s, blah blah blah… Aufbau’s Principle states that in the ground state of an atom or ion, electrons fill atomic orbitals of the lowest available energy levels before occupying higher levels. For example, the 1s subshell is filled before the 2s subshell is occupied. Hund’s Rule states that every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied, and all electrons in singly occupied orbitals have the same spin. Quantum numbers are numbers that describe the energy state of a particle. Pauli’s Exclusion Principle states that no two electrons can have the same set of quantum numbers. “n” is principal quantum number which describes the size and energy level of the orbitals, “l” is the angular momentum quantum number which describes the shape of the orbital, “ml” is the magnetic quantum number which describes the orientation of the orbital with respect to another orbital, and ms is electron spin (up arrows are +.5, down arrows are -.5) (4). It turns out that the principles of electron configuration are key in predicting the existence of bosons and fermions.

Classical Mechanics vs. Quantum Mechanics

Two particles are identical if their defining intrinsic properties are the same. All electrons are identical to each other, because they share the same intrinsic properties such as mass, charge, and spin. All muons are identical to each other because they also share the same intrinsic properties. However, electrons and muons are not identical because they have the same charge and the same spin, but they do not have the same mass. Identical-ness is not the same as quantum state. All electrons are identical, but they are in different states. For example, two electrons may have different momentum, or one might be bound to a proton while the other is not. Exchanging identical particles does not affect the properties of the system. This statement has vastly different ramifications in classical systems and quantum systems. Let’s name one particle “particle 1” and the other particle “particle 2.” In a classical system, identical particles are distinguishable. In classical mechanics, two identical particles are distinguishable because we are able to track them at every position as they move, but we cannot do that in quantum mechanics. Just like in classical mechanics, even though the particles are identical, their states are different; however, after we initially label the two particles, the particles interact. Because in a quantum system we cannot observe a quantum state there is no way of telling which particle is which after the initial moment that we name them (5).

Probability Distribution

In quantum mechanics, the wave function is a mathematical description of everything we know about a quantum system. The wave function is represented by the Greek letter Ψ. The wave function is very closely linked to the probability distribution of the system. The wave function squared results in the probability distribution (6). This probability distribution function represents the odds of finding a particle at a specific position. The value that the function produces increases and decreases depending on the value of r1 (our particle).

Predicting Bosons and Fermions Mathematically

In our original wave function, we determined the probability of the position of a particle in a one particle system. Let’s imagine a two particle system with “particle a” and “particle b” in which the two particles are indistinguishable. We can design a single wave squared function for the entire system.

Because the two particles are indistinguishable, then the odds of finding the particles in position (r1,r2) must be equal to the odds of finding the particles in the position (r2,r1), otherwise we would be able to distinguish between the two particles.

We can now split this result into two separate equations:

We have now determined two separate kinds of indistinguishable particles. Particles whose wave functions are equal to each other are bosons and particles whose wave functions are the opposite of each other are fermions.

Bosons vs. Fermions

Particles that behave like bosons include the Higgs Boson, photons, and gluons. Particles that behave like fermions include electrons and quarks. Bosons turn out to be able to perform Bose-Einstein Condensation. These types of particles do not follow Pauli's Exclusion Principle. Fermions such as electrons do follow Pauli's Exclusion Principle demonstrated by the fact that their wave functions are not equal. However, fermions are unable to perform Bose-Einstein Condensation.

Predicting Bose-Einstein Condensates

Let’s imagine a system with two Bosons. The particles can occupy two different energy levels: the 0 energy level which is the ground state of the particle and the 1 energy level is a slightly higher energy level. We can model possible wave functions for our system (7).

In model 1, particle a in energy level zero and particle b in energy level 0. This wave function is symmetric because it remains the same if we swap the positions of the particles. Because the two particles are Bosons, they do not follow Pauli’s exclusion principle and can occupy the same state.

Now let’s imagine a wave function for two fermions. The wave function must be antisymmetric. Under this model, the two particles are in a state of superposition. In quantum mechanics, superposition is the state of a particle before it is measured when it can occupy two states at the same time. However, as soon as we observe a particle in superposition, the superposition collapses and the particle must choose one singular state. In the third model, the bosons are also in a state of superposition; however, they are able to occupy the same state. The negative sign in the model of the fermions shows that the two particles cannot occupy the same state. Either particle a will be in energy level 1 and particle b will be in energy level 0, or particle a will be in energy level 0 and particle b will be in energy level 1. This difference relates to Bose-Einstein Condensation. In our example it is just a two particle system, but the bosons are able to occupy the same state, but the fermions are unable to occupy the same state. We can extend this model to a system with far more particles. Electrons are examples of fermions. Our model is supported by both Aufba’s Principle and Hund’s Rule about the way in which electrons arrange themselves in different energy levels. Two electrons can be in the same energy level; however, they cannot occupy the same energy level unless they have different spins. Bose-Einstein Condensation is able to occur because unlike fermions, bosons are able to all occupy the same energy state. If we cool bosons to a very cold temperature, all of the bosons will collapse into the one energy state, most likely the lowest energy state.

Sources

(1) https://www.livescience.com/65427-fundamental-elementary-particles.html

(2) https://www.pngkit.com/bigpic/u2e6u2q8t4o0e6i1/

(3) http://kaffee.50webs.com/Science/activities/Chem/Activity.Electron.Configuration.html

(4) https://www.youtube.com/watch?v=C6afrc1QS6Y

(5) https://www.youtube.com/watch?v=1cIl3m-fmnY

(6) https://www.youtube.com/watch?v=skFU7pmBOys

(7) https://www.youtube.com/watch?v=lBpxQdikm0w