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

Since the potential is infinite outside of the given region, the wavefunction of the particle must be zero at the boundaries of the box. This implies that the wavefunction must have nodes at the boundaries. The ground state wavefunction will have no nodes (it is just a constant) and thus has zero energy. The first excited state must have one node along each direction, and thus three nodes in total. This gives rise to three possible wavefunctions: one with a node along x, one with a node along y, and one with a node along z. These three wavefunctions have the same energy, and thus there is a triple degeneracy. The energy of the first excited state is therefore 2E₀ (twice the ground state energy).

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

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