An electron is in a three-dimensional box. The x- and...

An electron is in a three-dimensional box. The $x$- and $z$-sides of the box have the same length, but the $y$-side has a different length. The two lowest energy levels are 2.24 eV and 3.47 eV, and the degeneracy of each of these levels (including the degeneracy due to the electron spin) is two. (a) What are the $n_X, n_Y$, and $n_Z$ quantum numbers for each of these two levels? (b) What are the lengths $L_X, L_Y,$ and $L_Z$ for each side of the box? (c) What are the energy, the quantum numbers, and the degeneracy (including the spin degeneracy) for the next higher energy state?

Key Concepts

Particle in a Box

This concept describes a quantum system where a particle is confined within an impenetrable potential well. The confinement leads to discretized energy levels because the particle's wavefunction must satisfy specific boundary conditions. In three dimensions, the box's geometry plays a significant role, and when the dimensions are unequal (anisotropic), it creates an unequal spacing of energy levels along each axis.

Quantum Numbers

Quantum numbers (n?, n?, n_z) label the allowed stationary states of a particle in a box. They are positive integers that arise from solving the Schrödinger equation with boundary conditions. Each set of quantum numbers corresponds to a unique energy state, and the energy of the state is dependent on the squares of these numbers divided by the square of the corresponding side lengths.

Energy Quantization Formula

The quantization of energy in a particle in a box is given by an expression that sums the contributions from each spatial dimension: E ? (n?² / L?²) + (n?² / L?²) + (n_z² / L_z²). This formula demonstrates how the geometry of the box and the specific quantum numbers influence the total energy of the system.

Degeneracy

Degeneracy refers to the occurrence of two or more distinct quantum states sharing the same energy level. In the context of a 3D box, degeneracy may arise from different combinations of quantum numbers that lead to identical energy values, and additionally, the electron's intrinsic spin (typically two possible orientations) contributes to a two-fold degeneracy in each energy level.

Anisotropic Confinement

Anisotropic confinement occurs when the dimensions of the box are not equal. This results in different energy spacings along each axis, as the contribution to the total energy is inversely proportional to the square of the corresponding side length. In problems involving such geometries, identifying the order of energy levels requires careful consideration of these differences.

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