Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A1 / a

Inside the infinite square well, the stationary wavefunctions can be chosen real:They vanish outside the well and at its impenetrable endpoints. A sine meets the left boundary condition; the right one requires $\sin (n\pi )=0$. The putative $n=0$ function is identically zero, so it is not a quantum state. The constants $C_n$ are fixed by wavefunction normalization.

Wavefunction normalization requires unit total probability:Choosing an irrelevant overall phase to be zero givesThe amplitude has units of inverse square root length, so $|{\Psi }_n{|}^2\mathrm{d}x$ is dimensionless.

Within the well, the potential is zero and the Hamiltonian is purely kinetic:The infinite walls are encoded in the operator's boundary conditions $\Psi (0)=\Psi (a)=0$, rather than differentiated as part of the interior expression. Sincewe obtainThese positive energy eigenvalues grow quadratically with the quantum number; the ground state has a nonzero confinement energy.

For distinct positive integers $n,m$, use the product-to-sum identity in the inner product:Every endpoint sine vanishes. For $m=n$, the normalization integral is one; the expression with a denominator $m-n$ should not be substituted directly. Thus the normalized functions form an orthonormal set, $⟨{\Psi }_m|{\Psi }_n⟩={\delta }_{mn}$.

Project the expansion onto one normalized basis function. Orthogonality removes every other coefficient:Consequently,This is a Fourier sine series in normalized form. For a square-integrable $f$, completeness gives convergence in the norm of the function space; pointwise convergence requires the usual additional regularity. If $f$ is a normalized wavefunction, the Born rule gives probabilities $|a_n{|}^2$ and ${\sum }_n|a_n{|}^2=1$.