A particle is in a three-dimensional box. The y length of...

A particle is in a three-dimensional box. The $y$ length of the box is twice the $x$ length, and the $z$ length is one-third of the $y$ length. (a) What is the energy difference between the first excited level and the ground level? (b) Is the first excited level degenerate? (c) In terms of the $x$ length, where is the probability distribution the greatest in the lowest-energy level?

A particle is in a three-dimensional box. The $y$ length of the box is twice the $x$ length, and the $z$ length is one-third of the $y$ length. (a) What is the energy difference between the first excited level and the ground level?
(b) Is the first excited level degenerate? (c) In terms of the $x$ length, where is the probability distribution the greatest in the lowest-energy level?

Key Concepts

Particle in a Box

This is a foundational quantum mechanics problem where a particle is confined within impenetrable boundaries. The system’s allowed energy levels and wavefunctions are determined by solving the Schrödinger equation using boundary conditions that require the wavefunction to vanish at the edges of the box. This model illustrates the quantization of energy and spatial confinement effects intrinsic to quantum systems.

Quantized Energy Levels

In a box with finite dimensions, the energy levels of a particle depend on the square of the quantum numbers associated with each spatial dimension, as well as on the dimensions of the box. The energy formula arises from the separation of variables in the Schrödinger equation, and demonstrates that different configurations or dimensions scale the energy levels in distinct ways, leading to a discrete spectrum of allowed energies.

Degeneracy

Degeneracy refers to the occurrence of two or more distinct quantum states sharing the same energy level. In the context of a particle confined in a multi-dimensional box, particular sets of quantum numbers can yield identical energy values even if the individual quantum numbers differ. This concept is crucial when analyzing the state structure and understanding how modifications in geometric parameters affect the energy spectrum.

Probability Density Distribution

The probability density in quantum mechanics, given by the squared modulus of the wavefunction, represents the likelihood of finding a particle at a given point. For the ground state of a confined system, the wavefunction typically exhibits its maximum amplitude at the center of the box, but the precise location of the maximum probability can be influenced by the dimensions of the box and the boundary conditions imposed.

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