Imagine the universe to be a spherical cavity, with a radius of 10^28 cm and impenetrable walls

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Imagine the universe to be a spherical cavity, with a radius...

Imagine the universe to be a spherical cavity, with a radius of $10^{28} \mathrm{~cm}$ and impenetrable walls. (a) If the temperature inside the cavity is 3 K , estimate the total number of photons in the universe, and the energy content in these photons. (b) If the temperature were 0 K , and the universe contained $10^{80}$ electrons in a Fermi distribution, calculate the Fermi momentum of the electrons.

Imagine the universe to be a spherical cavity, with a radius of $10^{28} \mathrm{~cm}$ and impenetrable walls.
(a) If the temperature inside the cavity is 3 K , estimate the total number of photons in the universe, and the energy content in these photons.
(b) If the temperature were 0 K , and the universe contained $10^{80}$ electrons in a Fermi distribution, calculate the Fermi momentum of the electrons.

Key Concepts

Degenerate Fermi Gas

At absolute zero temperature, fermionic particles (like electrons) fill up the lowest available quantum states up to a maximum energy known as the Fermi energy. This framework is essential for understanding the behavior of electrons in highly dense systems, where quantum statistics and the Pauli exclusion principle are key.

Fermi Momentum

Fermi momentum is defined as the momentum corresponding to the highest occupied quantum state in a degenerate Fermi gas at zero temperature. It is directly related to the number density of fermions through quantum statistical mechanics and is important for predicting the physical properties of systems consisting of many fermions under degenerate conditions.

Volume of a Sphere

The volume of a spherical region is crucial when converting density (per unit volume) into a total quantity. This geometric concept is used to calculate how large-scale quantities, such as total photon number or energy, scale with the size of the universe or any spherical cavity.

Photon Number Density and Energy Density

These concepts involve the determination of the number of photons per unit volume and the total energy per unit volume emitted by a blackbody at a given temperature. Integrating the Planck spectrum yields the number density and energy density, which can then be scaled by the volume of a system to obtain the total number of photons and their collective energy.

Blackbody Radiation

This concept refers to the electromagnetic radiation emitted by an idealized object that absorbs and re-emits all incident radiation. The radiation spectrum depends solely on temperature and is described by Planck's law, which is fundamental in determining characteristics like photon number density and energy distribution at thermal equilibrium.

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