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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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Requisitos previos obligatorios, hasta el fondo: 43
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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.
Errores frecuentes
- Treating unbounded operators as if they were bounded; their domains matter.
- Interpreting the uncertainty principle only as a statement about clumsy measurement.
Conduce a
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Itinerarios de aprendizaje
Lecturas complementarias
- Brian C. Hall, Quantum Theory for Mathematicians (2013)
- Leon A. Takhtajan, Quantum Mechanics for Mathematicians (2008)