Quantum Week: Pauli's Exclusion Principle And Its Wonderful Applications In Astronomy.

On the 5th day of the quantum week, we'll talk about a famous principle of quantum mechanics that was put forward by Wolfgang Pauli: The Exclusion Principle. We'll also learn how this principle comes into picture in astrophysics and saves a dying star.

Pauli's exclusion principle is simple to understand. Let us first understand it using an analogy. Suppose you have two friends A and B studying in an University. You want to give them a surprise visit. You go to their hostel. They are in the same hostel. Let us say the hostel has two blocks. A and B are in the same block. So you enter that block. Next, you ask for their rooms. They are in the same room.  Now when you open the room, they are sleeping. Here is the point: The room has two single beds. A and B will always be sleeping on their own respective beds. They will not be in the same single bed. This is what the Exclusion Principle means. Now let us relate our analogy to the quantum world.

Suppose you want to know the positions of two electrons. Let them be in the same shell, say the second shell (principle quantum number n=2). You go deeper and find they are in the same orbital, say, the p orbital. (angular momentum quantum number l = 1). Further, let them have the same magnetic quantum number m. So far so good. But now if you finally reach the deepest level, you'll see that the two electrons will never have the same spin quantum number s. So in the last part, the two electrons will have different orientation. When we compare it with our analogy, we see that n is the hostel, l is the block, m is the room and s are the beds. The two cannot be in the same bed/quantum state.

2 fermions (electrons) cannot occupy the same quantum state with all the quantum numbers identical.

All the particles in the universe are divided into two categories: Fermions, having half integral (1/2,3/2 etc.) spin and Bosons, having integral (1,2,3 etc.) spin. The exclusion principle is obeyed only by the Fermions. Why? We don't know. A rigorous reason given by a quantum physicist will be that, 'the wavefunction of a Fermion is anti-symmetric under the exchange operator'. But that is a mathematical reason. It is not a physical reason.

Now in order to realize the wonderful application of Pauli's exclusion principle in astrophysics, we first need to answer two important questions: How do stars evolve and how does Pauli’s exclusion principle save the star. We begin with the first question: How do stars evolve? I will explain it in the simplest and briefest possible way.

A star is a hot ball of plasma. There is a core region in the star that hosts nuclear fusion reaction. A star spends 90% of its life fusing the most basic nuclear fusion reaction: hydrogen to helium in its core. Such  a star is known as the main sequence star, an example is our Sun. The main characteristic of such a star is that it is in perfect hydrostatic equilibrium. Getting complicated? Consider this: The star is massive. So massive that it starts collapsing under its own gravity. But what stops the inward gravitational collapse is the outward (gas) pressure of the core nuclear reaction. So the inward gravitational collapse is perfectly balanced by the outward gas pressure and such a star is said to be in hydrostatic equilibrium.

When all the hydrogen is converted into helium, next element in the chain, carbon, forms. The temperature required for the hydrogen fusion was about 15 million K and for helium fusion to carbon is about 100 million K. One day, again, all the helium burns out into carbon and what is left is an inert carbon core. The temperature required to fuse carbon is whooping 500 million K. Small to mid sized stars do not have the potential to host a full scale carbon fusion. That is all we need to know about stellar evolution to understand the application of Pauli's principle. We now move on to the next question.

The Life Cycle of Stars.

Now in the absence of the core reaction, gravity gains the upper hand and starts collapsing the star. This collapse starts increasing the density of the core region. Thus the mean separation between the constituent particles decreases and becomes of the order of de-Broglie wavelength (ignore this wavelength concept if its difficult to understand. Just learn that the density increases and separation b/w particles decreases). Such a system of high density is known as degenerate system. Now who will the save the star?

Answer is electrons! These little sub-atomic particles hate being crushed. They are fermions and obey Pauli’s exclusion principle. Thus no more than two electrons (one with spin up and other spin down) can occupy the same quantum state. So as gravity tries to crush the star, all the available lower energy states start getting filled. (See Fermi Energy to understand it deeply). Now since other electrons cannot occupy the already filled lower energy states, they have to fill the higher energy states. Lower energy state electrons will say, “No! You cannot occupy this state. We have already occupied it. You need to go to higher energy levels. We will exert an outward pressure if you try to occupy this state.” This pressure is known as the electron degeneracy pressure. Thus, in a highly degenerate system, the electrons with highest energy states have incredible amount of velocity associated with them (because they are at higher energy levels and hence kinetic energy is high). So high, that now relativistic effects come into picture. (For complete mathematics of this concept, see this).

In a stable white dwarf star, the inward gravitational collapse is balanced by this electron degeneracy pressure. But if the mass of the star becomes more than 1.4 solar masses, even the electron degeneracy pressure will break down. The electrons will then combine with protons and form neutrons and thus, a neutron star. This mass limit, below which the white dwarf star is stable, is known as the Chandrasekhar limit.

Size comparison of Sirius B (nearest white dwarf) with Earth.

Pauli's principle and Fermi energy are the core concepts of quantum statistical mechanics. To study astrophysics, or to be an astrophysicist, quantum mechanics and statistical mechanics are very important.
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