Self-adjointness and physical boundaries

For a densely defined operator A, symmetry means its inner-product boundary form vanishes for pairs of vectors in its chosen domain. Self-adjointness requires more: the adjoint must have exactly that same domain and action. An unbounded operator cannot generally act on every Hilbert-space vector. The adjoint domain consists of vectors for which the relevant inner-product functional is bounded, so it depends on both the expression and the original domain.

Consider momentum on a finite interval. Integrating its first derivative by parts leaves endpoint values. Setting both endpoint values to zero certainly removes this form for vectors in the original domain, making the operator symmetric. But testing against such functions imposes no endpoint restriction on an adjoint vector. The adjoint domain is then larger, so this two-endpoint Dirichlet realization of momentum is not self-adjoint. A real expectation value on selected vectors does not repair that mismatch.

A quasiperiodic condition links the endpoint values by one fixed unit-modulus phase. For the first derivative, the domain of H¹ functions satisfying this relation yields a self-adjoint momentum realization. Both endpoint contributions cancel because bra and ket obey the same phase condition, and the adjoint condition enforces that same relation. Different phases specify different operators, even though their derivative formula is identical.

The eigenfunctions are plane waves. Requiring the phase accumulated across the interval to match the domain phase quantizes the momentum. Periodic boundary conditions allow zero momentum; antiperiodic boundary conditions shift the whole integer ladder by half a step. This model describes a circle with an endpoint phase identification, not a particle bouncing off two impenetrable walls. For the free Hamiltonian $P_\theta^2/(2m)$, the induced $H^2$ domain also requires $\psi'(L)=e^{i\theta}\psi'(0)$. Together the value and derivative conditions make Schrödinger currents agree at the identified endpoints, allowing circulation without loss.

The kinetic Hamiltonian uses a second derivative and has a different boundary form involving values and first derivatives. Its Dirichlet realization on an interval can be self-adjoint even though the first derivative with Dirichlet conditions at both ends is not. Squaring a formal expression and assigning unrelated boundary conditions is therefore not the same as taking the operator square with its induced domain. This distinction explains many apparent contradictions about momentum inside an infinite well.

A self-adjoint Hamiltonian generates unitary time evolution through the spectral theorem. Merely demonstrating a formally conserved norm for smooth trial functions does not establish that the full initial-value problem has the required generator. In applications, identify the geometry and boundary physics, specify an appropriate self-adjoint realization, and only then use its eigenbasis or evolution. Not every mathematically allowed extension represents the same apparatus, and a boundary phase is physical input rather than a freely adjustable normalization constant.