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

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.

Twice differentiation the sine givesBoth Dirichlet boundary conditions hold when $n$ is an integer. The zero function at $n=0$ is not an eigenfunction, and negative $n$ merely changes an overall sign. ThereforeThe corresponding normalized eigenfunction is $\sqrt{2/a}\sin (n\pi x/a)$.