The Pauli Exclusion Principle has profound implications for the entire universe. However, this principle is not itself a fundamental law of nature, but is based on a much deeper fundamental principle which is even more profound. Suppose we have one particle trapped inside a one-dimensional box in the lowest possible energy state. Its wave function will look like this. The amplitude of the wave function depends on both the real component and the imaginary component. If we add energy to the particle, the frequency of the wavefunction will increase. If we add more energy, the frequency will increase even more. Now suppose we have two identical particles trapped inside a box, and suppose that the two particles do not exert any forces on each other. The variable X1 will represent the position of the first particle and X2 will represent the position of the second particle. Although we are dealing with only one spatial dimension, we can create a two-dimensional graph of every possible combination of X1 and X2. We can create a two-dimensional graph of every possible combination of X1 and X2. When dealing with quantum systems involving multiple particles, each particle does not have its own wavefunction, but rather there is a single wavefunction representing all the particles. Here, the height represents the real component, and the color represents the imaginary component. Both the real and imaginary components contribute to the amplitude. The probability of observing the particles at any particular combination of X1 and X2 is given by the square of the amplitude of the wavefunction at X1 and X2. Let's consider a new scenario, where we have two identical particles, each trapped in its own box. It is a fundamental property of the universe that identical particles are indistinguishable. In other words, we can’t tell which particle is represented by X1 and which particle is represented by X2. This means that the probability that particle 1 will be at position X1, and that particle 2 will be at position X2, must be exactly equal to the probability that particle 1 will be at position X2, and that particle 2 will be at position X1. If both particles are in their lowest possible energy states, we get the following amplitude, with the probability being equal to the square of the amplitude. There is more than one wavefunction that can create this amplitude distribution. We can have a symmetric total wavefunction, as is the case for particles such as photons. Or we can have an anti-symmetric total wavefunction, as is the case for particles such as electrons. The total wavefunction consists of both the spatial wavefunction and the spin. For now, let’s just focus on the spatial component of the wavefunction. Let’s place both particles in the same box, and ignore all the forces that the particles exert on each other. If we have a symmetric spatial wavefunction, then the two parts of the wavefunction will reinforce each other. We therefore have no problem having many identical particles together in the same quantum state if they have symmetric wavefunctions, as photons do. On the other hand, if we have an anti-symmetric spatial wavefunction, then when we place the particles together in the same box, the two parts of the wavefunction will cancel each other out, and we will get a probability of zero everywhere. In other words, particles can never be together in the same quantum state if they have anti-symmetric spatial wave functions. This is what creates the Pauli Exclusion principle for particles such as electrons, but the most beautiful part of the story is still yet to come. Suppose that we again start out with two particles in separate boxes, but this time one of the particles has more energy than the other. As before, the particles are indistinguishable, so we don’t know which particle is described by X1 and which particle is described by X2. This is the waveform for an individual particle in the lower energy level. And this is the waveform for an individual particle in the higher energy level. This is both waveforms together. Since we don’t know which particle is described by X1 and which particle is described by X2, we get two variations for each individual waveform. Since we don’t know which particle is described by X1 and which particle is described by X2, we get two variations for each individual waveform. Let's multiply these waveforms together. Here, we have the symmetric wavefunction for two identical particles in two separate boxes, in two different energy states. Let's now place both particles in the same box, and ignore all the forces between the two particles. This is how it will look for the case of a symmetric spatial wavefunction. Let's now repeat this again, but this time with an anti-symmetric spatial wavefunction. Note that we now have a negative sign on the right. Let’s now place both particles in the same box, and again ignore all the forces between the two particles. Even though this is an anti-symmetric spatial wavefunction, it is now possible for the two particles to be in the same box, because they have different energy levels. Notice how the amplitude is zero everywhere along the line where X1 is equal to X2. This means that there is a zero probability of the particles being at the same location, and this creates an apparent repulsion between the two particles. Keep in mind that we are ignoring all the forces between the two particles, yet we are getting an apparent repulsion anyway. On the other hand, consider the case for a symmetric spatial wavefunction. Here, we have the opposite situation, in that we have a higher amplitude along the line where X1 is equal to X2. This means that the particles are more likely to be near each other, and we have an apparent attractive force. So, we have an apparent attractive force for symmetric spatial wavefunctions, and an apparent repulsive force for the anti-symmetric spatial wavefunctions. Whether the "total" wavefunction is symmetric or anti-symmetric depends both on the spatial wavefunction and on the spin. Photons fall into the category of particles we call "bosons", which always have symmetric total wavefunctions. This means that when we swap the labels of X1 and X2, the total wave function stays the same. Electrons fall into the category of particles we call "fermions", which always have anti-symmetric total wavefunctions. This means that when we swap the labels of X1 and X2, the total wave function is multiplied by negative one. Let us suppose we are dealing with two electrons, and let us ignore the electromagnetic forces. There are four different spin combinations. There are four different spin combinations. We can think of each spin combination as having its own spatial wavefunction. In this example, both particles are spinning up. In this other example, both particles are spinning down. In this example, one particle is spinning up and the other particle is spinning down. Since the particles are indistinguishable, we don’t know which particle is which, and we get a combination of these two outcomes as shown. Electrons, and other fermions, always have anti-symmetric total wave functions, meaning that if we swap the labels of X1 and X2, the total wavefunction is multiplied by negative one. Here is another way that the total wave function can be anti-symmetric. The plus sign has changed to a minus sign. In this case, we say that the spin is anti-symmetric. Notice that the spatial wavefunction is now symmetric. Electrons, and other fermions, can have symmetric spatial wave functions if their spins are anti-symmetric. A symmetric spatial wave function is required for the two electrons to be together in the lowest energy level, and this is allowed if the spins are anti-symmetric. Let's separate the two particles into different boxes, and increase the difference in energy levels between the two particles even further than before. Let's now place the particles in the same box again, and again ignore all the forces between them. We get this result for the symmetric spatial wave function. Now, let's repeat this, but with the anti-symmetric version of the spatial wavefunction. Let's now try a different combination of energy levels. Here we have the symmetric spatial wavefunction. Here we have the anti-symmetric spatial wave function. Now let's consider an example where the two energy levels are equal. In the case of the symmetric spatial wave function, the two waveforms strengthen each other. In the case of the anti-symmetric spatial wave functions, the two waveforms cancel each other out, yielding a zero probability everywhere, implying that this can never happen. The fact that the wavefunctions cancel each other out for anti-symmetric spatial wave functions is what causes the Pauli Exclusion Principle. The Pauli Exclusion Principle is what prevents all the electrons of every atom from all simultaneously falling to the lowest energy level. Without the Pauli Exclusion Principle, there would be no chemistry and no life. Much more information is available in the other videos on this channel. Please subscribe for notifications when new videos are ready. And if you are able to, please consider supporting us on Patreon through the link in the video description. Thank you.
