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White dwarf stars are the remnants left after a star like our Sun has exhausted its nuclear fuel and shed its outer layers. They are incredibly dense objects typically the size of Earth but contain the mass of about the Sun. Unlike main sequence stars, white dwarfs do not undergo nuclear reactions; instead they shine due to the residual thermal energy left from their previous life cycle.

As final endpoints of stellar evolution for medium-sized stars, white dwarfs pack a significant amount of mass into a very small volume, resulting in extreme density. This leads to a unique state of matter not found on Earth where the principles of quantum mechanics dominate, giving rise to a phenomenon known as electron degeneracy pressure. This pressure is critical to a white dwarf's stability and dictates that no two electrons can occupy the same energy state simultaneously.

White dwarf stars gradually cool and fade away over time, but their evolution takes yet another turn if they begin to accrete matter from their surroundings, which can lead to surpassing a critical mass limit known as the Chandrasekhar limit.

Electron degeneracy pressure is a quantum mechanical effect that arises in compact objects like white dwarf stars. When such a star has burnt out its nuclear fuel, it no longer supports itself through the force of radiation pressure from nuclear reactions happening at its core. Instead, it relies on electron degeneracy pressure to counteract the force of gravity.

This pressure comes from the Pauli exclusion principle, which states that no two fermions (such as electrons) can occupy the same quantum state. In a white dwarf, electrons are packed so densely that further compression would force them into the same energy state. This is forbidden by quantum mechanics, resulting in a pressure that resists further compression. It is this pressure that holds up a white dwarf against gravitational collapse.

However, there is a limit to how much mass electron degeneracy pressure can support, leading to the concept of the Chandrasekhar limit. If a white dwarf exceeds this mass, it can no longer support itself with electron degeneracy pressure alone and undergoes further gravitational collapse.

Stellar evolution is the process by which a star changes over time, governed by the delicate balance between the gravitational force and the pressure from within. For stars not exceeding the Chandrasekhar limit, the final stage is as a white dwarf. However, when a white dwarf's mass goes over this threshold, it faces a different destiny, unable to maintain stability through electron degeneracy pressure.

A white dwarf that accumulates additional mass from a companion star, for instance, may exceed the Chandrasekhar limit and trigger a Type Ia supernova, an explosive event that plays a critical role in dispersing heavy elements into space. On the other hand, if the star was initially more massive, the post-white dwarf evolution could lead to the formation of a neutron star or, in the case of the most massive stars, a black hole.

Each pathway in stellar evolution is a piece of the cosmic puzzle, shedding light on the lifecycle of stars and the chemical evolution of galaxies. The understanding of these processes also aids in the study of deep cosmological questions like the rate of expansion of the universe, as Type Ia supernovae serve as standard candles for astronomical measurements.