Introduction: Eigenstates Are Perpendicular
The infinite well eigenstates are orthonormal—mutually perpendicular in Hilbert space. This is guaranteed by the spectral theorem since the Hamiltonian is Hermitian.
The Orthonormality Relation
\[\int_0^L \psi_m^*(x)\psi_n(x)\, dx = \delta_{mn} = \begin{cases} 1 & m = n \\ 0 & m \neq n \end{cases}\]Verification
\[\int_0^L \frac{2}{L}\sin\left(\frac{m\pi x}{L}\right)\sin\left(\frac{n\pi x}{L}\right) dx\]Use the identity: \(2\sin A\sin B = \cos(A-B) - \cos(A+B)\)
The integral gives 0 for \(m \neq n\) and \(L/2\) for \(m = n\).
With normalization factor \(\sqrt{2/L}\): result is \(\delta_{mn}\). ✓
Completeness
The eigenstates form a complete basis. Any function on \([0, L]\) satisfying the boundary conditions can be expanded:
\[f(x) = \sum_{n=1}^{\infty} c_n \psi_n(x)\]This is a Fourier sine series!
The Quantum Connection
Orthonormality ensures that expansion coefficients \(c_n = \langle\psi_n|f\rangle\) give unique, well-defined probabilities \(|c_n|^2\). Different energy states are completely distinguishable—measuring energy collapses the state to exactly one eigenstate.