For a particle in a three-dimensional cubical box, what is...

For a particle in a three-dimensional cubical box, what is the degeneracy (number of different quantum states with the same energy) of the energy levels
(a) $3 \pi^{2} \hbar^{2} / 2 m L^{2}$ and
(b) $9 \pi^{2} \hbar^{2} / 2 m L^{2} ?$

Key Concepts

Particle in a Cubical Box

This concept deals with the quantum mechanical problem where a particle is confined in a three-dimensional box with infinitely high potential walls. The confinement leads to a quantization of energy since the particle’s wavefunction must satisfy specific boundary conditions, making the model a fundamental example in quantum mechanics used to illustrate spatial quantization and the behavior of electrons in confined geometries.

Energy Quantization

Energy quantization in a confined system means that the allowed energies are not continuous but take on discrete values. For a particle in a three-dimensional box, the energy depends on the square of three quantum numbers, corresponding to each spatial dimension. This quantization results from solving the Schrödinger equation with the appropriate boundary conditions, yielding an expression where the energy is proportional to the sum of the squares of these quantum numbers.

Quantum Degeneracy

Degeneracy refers to the phenomenon where different quantum states (distinct sets of quantum numbers) yield the same energy level. In three-dimensional systems, various combinations of quantum numbers can produce the same sum of squares, leading to multiple states sharing identical energy. The concept of degeneracy is important because it influences the properties of systems such as spectral lines, statistical mechanics calculations, and how systems respond to external perturbations.

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