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"canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-21-01/",
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"id": "PHYS-2026-07-21-01",
"date": "2026-07-21",
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"title": "Self-adjointness as boundary data",
"summary": "Why choosing the domain of a quantum Hamiltonian is part of specifying the observable.",
"language": "en",
"entry_kind": "daily",
"status": "published",
"level": "graduate-advanced",
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"content_markdown": "\n## Key claim\n\nA differential expression is not yet an observable. The fracture appears when the same formal Hamiltonian acquires different spectra because its domain changes.\n\n## Theme\n\nSelf-adjointness is boundary data in disguise.\n\n## Guiding question\n\nFor a particle on the interval $[0,L]$, suppose we write\n\n$$\nH=-\\frac{\\hbar^2}{2m}\\frac{\\dd^2}{\\dd x^2}.\n$$\n\nWhat physical information is missing from this line?\n\n## Setup\n\nIntegration by parts exposes the boundary form\n\n$$\n\\bra{\\phi}H\\ket{\\psi}-\\bra{H\\phi}\\ket{\\psi}\n=-\\frac{\\hbar^2}{2m}\n\\left[\\phi^*(x)\\psi'(x)-\\phi'^*(x)\\psi(x)\\right]_{0}^{L}.\n$$\n\nSymmetry requires this expression to vanish on the chosen domain. Self-adjointness further requires that the adjoint have exactly the same domain.\n\n## Analysis\n\n### Derivation\n\nBuild a domain by imposing Dirichlet data, $\\psi(0)=\\psi(L)=0$. The boundary form vanishes, and the familiar discrete spectrum follows.\n\n### Scope and assumptions\n\nWhy privilege Dirichlet data? Periodic data,\n\n$$\n\\psi(L)=e^{\\ii\\theta}\\psi(0),\\qquad\n\\psi'(L)=e^{\\ii\\theta}\\psi'(0),\n$$\n\nalso makes the boundary form vanish and defines a different self-adjoint operator.\n\n### Physical interpretation\n\nThe formal differential rule describes local evolution. The domain tells the wavefunction how the ends of configuration space are physically identified.\n\n## False claim to diagnose\n\n> If two Hamiltonians have the same differential expression, they represent the same observable.\n\nLocate the hidden assumption that makes this statement false.\n\n## What follows — and what does not\n\nThe Builder treats the boundary condition as a construction choice. The Skeptic reveals a family of equally consistent choices. The Translator identifies the choice as global physical structure rather than a mathematical afterthought.\n\n## Exercise\n\nLet $H=-\\dd^2/\\dd x^2$ on $[0,L]$. Show that Robin conditions\n\n$$\n\\psi'(0)=\\alpha\\psi(0),\\qquad\n\\psi'(L)=\\beta\\psi(L),\n$$\n\nwith real $\\alpha,\\beta$, make the boundary form vanish for every pair of functions satisfying the same conditions.\n\n\nHint 1
\n\nEvaluate the boundary form separately at $0$ and $L$.\n\n \n\n\nHint 2
\n\nUse the reality of $\\alpha$ and $\\beta$ when complex-conjugating the condition on $\\phi$.\n\n \n\n**Oral check 1.** Why would complex $\\alpha$ generally spoil symmetry?\n\n**Oral check 2.** Does vanishing of the boundary form alone prove self-adjointness, or only symmetry?\n\n\nSolution
\n\nAt $x=0$,\n\n$$\n\\phi^*(0)\\psi'(0)-\\phi'^*(0)\\psi(0)\n=\\alpha\\phi^*(0)\\psi(0)-\\alpha\\phi^*(0)\\psi(0)=0.\n$$\n\nThe same cancellation holds at $L$ with $\\beta$. Thus the boundary form vanishes. Establishing self-adjointness also requires verifying that the adjoint domain introduces no additional boundary freedom.\n\n \n\n## Check your understanding\n\nIn one sentence: what extra datum turns a formal Hamiltonian into a quantum observable?\n\n## Connections and next step\n\n- New pressure point: operator domain versus differential expression.\n- Retrieve later: deficiency indices and the $U(2)$ family of interval extensions.\n- Unresolved: which extensions can be generated as limits of regular boundary potentials?\n",
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