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Neutron Degeneracy Pressure

Pauli and his Principle

The Pauli exclusion principle is something that we all encounter on a daily basis. It is the law of the universe that prevents your laptop from falling through your lap. Formally, it states that two fermions (the particles that make up most matter) cannot occupy the same quantum states. More informally, two things cannot be in the same place, at the same energy level, with the same spin, at the same time. In neutron stars, this means that when mere heat is insufficient to halt collapse, the particles physically bumping into each other halts it instead.

This is called neutron degeneracy pressure. To understand it more thoroughly, we must introduce another principle: Heisenberg uncertainty.

Heisenberg's Uncertainty Principle

Werner Heisenberg, Nobel Laureate. In addition to the principle that is his namesake, he made many other contributions to quantum mechanics, such as in the problem of plasma physics. He also participated in the effort to develop a fundamental theory of particles.

Formally, the Heisenberg Uncertainty Principle states:

\[\Delta x \Delta p \ge \frac{\hbar}{2}\]

Let's break that down. \(\Delta x\) refers to the uncertainty in position, which you can think of as how many places the particle could be in. \(\Delta p\) refers to the uncertainty in momentum, which you can also think of as the range of values that its momentum (velocity times mass) could take. \(\hbar\) is known as the reduced Planck constant, and being a constant, is irrelevant to our understanding here.

Inside Neutron Stars

So how is this relevant? In neutron stars, where particles are compacted extremely close together, \(\Delta x\) is very small. Correspondingly, this means that \(\Delta p\) must be very large. Because of this, the average kinetic energy of the particles must be very large. (Kinetic Energy is related to momentum by \(KE =\frac{p^2}{2m}\))

We can see, then, that if high compression necessitates low position uncertainty, and low position uncertainty means high momentum, and high momentum means high energy, a tremendous amount of energy is required to squeeze neutron stars any further. This manifests itself as an outward pressure of sorts, referred to as degeneracy pressure. Matter that is stopped from being crushed mainly by degeneracy pressure, such as the plasma in neutron stars, is known as degenerate matter.

References

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[1] Atomic Archive: https://www.atomicarchive.com/resources/biographies/heisenberg.html