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The Lumenary
السنة الأولى من الدراسات العلياApplied Mathematicsمُخطَّط له

Mathematical foundations of quantum mechanics

Quantum mechanics in the language of functional analysis: states as unit vectors in Hilbert space, observables as self-adjoint operators, the spectral theorem, the Schrödinger equation and unitary evolution, and the harmonic oscillator.

math.applied.quantum-mechanics

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Functional analysis

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 43

ستتعلّم

  • Explain why observables are modelled by self-adjoint, possibly unbounded, operators and measurement outcomes by their spectra.
  • Solve the quantum harmonic oscillator using ladder operators.
  • Derive the Robertson–Heisenberg uncertainty relation from commutators.

تعريف

  • A pure state is a unit vector in a Hilbert space H, determined up to a phase.
  • An observable is a self-adjoint operator A on H; measuring it gives values in the spectrum of A.
  • Time evolution is ψ(t) = e^(−itH/ℏ) ψ(0), where H is the Hamiltonian.

مبرهنة

  • Stone's theorem: strongly continuous one-parameter unitary groups correspond to self-adjoint generators.
  • Robertson's inequality: σ_A σ_B ≥ ½ |⟨[A, B]⟩|.
  • The harmonic oscillator has energy levels ℏω(n + ½) for n = 0, 1, 2, ….
  • The canonical commutation relation [x, p] = iℏ has no solution in bounded operators.

مفاهيم خاطئة شائعة

  • Treating unbounded operators as if they were bounded; their domains matter.
  • Interpreting the uncertainty principle only as a statement about clumsy measurement.

اكتشافات اختيارية

قراءات إضافية

  • Brian C. Hall, Quantum Theory for Mathematicians (2013)
  • Leon A. Takhtajan, Quantum Mechanics for Mathematicians (2008)