A three-dimensional potential well has potential U₀= 0 in the region ,and and infinite potential otherwise.The ground state energy of a particle in the well is E₀.What is the energy of the first excited state,and what is the degeneracy of that state?

Question

Quizplus · Accepted Answer

In a three-dimensional potential well with a bound particle, the ground-state energy is given by: E1= 3h^2/(8mL^2), where L is the length of the cube. The first excited state will occur when the particle is excited to the first excited state in one of the dimensions, say the x-dimension. Therefore, the wavefunction will have two nodes along the x-dimension, and one node along each of the y and z-dimensions. The energy of the first excited state can be calculated using the formula: E2= n^2*h^2/(8mL^2), where n=2 (because there is one node along the x-dimension) and L is the length of the cube. Therefore, E2= (4/3)E1. Since there is only one possible way to excite the particle to the first excited state, the degeneracy of this state is single. Hence, the correct choice is (B) (4/3)E₀, single degeneracy.

A Three-Dimensional Potential Well Has Potential U0 = 0 in the Region

A Three-Dimensional Potential Well Has Potential U₀= 0 in the Region