Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 1 / A2 / b / i / Solution

Inside the particle in a box, the Quantum Hamiltonian isThe infinite potential outside imposes the Dirichlet boundary conditions $\psi (0)=\psi (a)=0$ and a zero wavefunction outside the interval. These boundary conditions are part of the Hamiltonian operator, rather than an optional addition.