A Modern Approach to Quantum Mechanics · Problem 10.10
Time-Independent Perturbations — John S. Townsend
Physics — Quantum Mechanics · cubic-infinite-potential-well
Problem
Determine the ground-state energy of a particle of mass in the cubic potential well Compare the volume of this infinite well with the spherical one (10.64) and discuss in general terms whether the relative values of the ground-state energies for the two wells are consistent with the position-momentum uncertainty relation.Intuitive explanation
A particle confined to a cubic box behaves like three independent 1D infinite wells—one for each direction. The ground state has the lowest quantum number in each dimension, giving you an energy that's the sum of three contributions. To compare with a spherical well of the same volume, think about how the geometry affects confinement: a cube concentrates the particle differently than a sphere, which changes the effective "tightness" of the potential and thus the zero-point energy. The uncertainty principle predicts that tighter spatial confinement forces larger momentum uncertainty, so whichever shape confines more sharply should have higher ground-state energy.
Intuitive explanation
A particle confined to a cubic box behaves like three independent 1D infinite wells—one for each direction. The ground state has the lowest quantum number in each dimension, giving you an energy that's the sum of three contributions. To compare with a spherical well of the same volume, think about how the geometry affects confinement: a cube concentrates the particle differently than a sphere, which changes the effective "tightness" of the potential and thus the zero-point energy. The uncertainty principle predicts that tighter spatial confinement forces larger momentum uncertainty, so whichever shape confines more sharply should have higher ground-state energy.