A particle is confined in a three-dimensional box with L₁= L₂ = 3L₃.The quantum numbers for the second excited state are

Question

Quizplus · Accepted Answer

The quantum numbers for the states of a particle in a three-dimensional box are given by (n_x, n_y, n_z), where n_x, n_y, and n_z are the quantum numbers corresponding to the x, y, and z dimensions, respectively. The energy of a state is proportional to the sum of the squares of these quantum numbers. The ground state has the quantum numbers (1,1,1), with each quantum number being the lowest possible value, 1. The first excited state increases one of these quantum numbers by 1, leading to states like (2,1,1), (1,2,1), or (1,1,2). For the second excited state, we can either increase one of the quantum numbers in the first excited state by one again or increase a different quantum number in the ground state to 2, leading to configurations like (1,2,1) and (2,1,1), which are indeed higher in energy than the ground state but lower than any state where the sum of the squares of the quantum numbers is greater. This makes option B correct, as it represents valid configurations for the second excited state, considering the lowest possible increases in quantum numbers from the ground state configuration.

A Particle Is Confined in a Three-Dimensional Box with L1

A Particle Is Confined in a Three-Dimensional Box with L₁