000 07591cam a2200373 i 4500
001 17686255
003 OSt
005 20220803123116.0
008 130405t20132013nyua b 001 0 eng c
010 _a 2013937175
020 _a9781461471158 (acidfree paper)
035 _a(OCoLC)ocn828487961
040 _aBTCTA
_beng
_cIISERB
_erda
042 _apcc
050 0 0 _aQC174.12
_b.H346 2013
072 7 _aQA
_2lcco
082 _223
_a530.12 H174Q
100 1 _aHall, Brian C.
_928450
245 1 0 _aQuantum theory for mathematicians
_cBrian C. Hall.
260 _aNew York:
_bSpringer,
_c2013.
300 _axvi, 554 pages :
_billustrations ;
_c24 cm.
490 1 _aGraduate texts in mathematics,
_x0072-5285 ;
_v267
504 _aIncludes bibliographical references (pages 545-548) and index.
505 0 0 _tThe experimental origins of quantum mechanics:
_tIs light a wave or a particle? ;
_tIs an electron a wave or a particle? ;
_tSchrödinger and Heisenberg ;
_tA matter of interpretation ;
_gExercises --
_tA first approach to classical mechanics:
_tMotion in R¹ ;
_tMotion in R[superscript n] ;
_tSystems of particles ;
_tAngular momentum ;
_tPoisson brackets and Hamiltonian mechanics ;
_tThe Kepler problem and the Runge-Lenz vector ;
_gExercises --
_tFirst approach to quantum mechanics:
_tWaves, particles, and probabilities ;
_tA few words about operators and their adjoints ;
_tPosition and the position operator ;
_tMomentum and the momentum operator ;
_tThe position and momentum operators ;
_tAxioms of quantum mechanics : operators and measurements ;
_tTime-evolution in quantum theory ;
_tThe Heisenberg picture ;
_tExample : a particle in a box ;
_tQuantum mechanics for a particle in R [superscript n] ;
_tSystems of multiple particles ;
_tPhysics notation ;
_gExercises --
_tThe free Schrödinger equation:
_tSolution by means of the Fourier transform ;
_tSolution as a convolution ;
_tPropagation of the wave packet : first approach ;
_tPropagation of the wave packet : second approach ;
_tSpread of the wave packet ;
_gExercises --
_tParticle in a square well:
_tThe time-independent Schrödinger equation ;
_tDomain questions and the matching conditions ;
_tFinding square-integrable solutions ;
_tTunneling and the classically forbidden region ;
_tDiscrete and continuous spectrum ;
_gExercises --
_tPerspectives on the spectral theorem:
_tThe difficulties with the infinite-dimensional case ;
_tThe goals of spectral theory ;
_tA guide to reading ;
_tThe position operator ;
_tMultiplication operators ;
_tThe momentum operator --
_tThe spectral theorem for bounded self-adjoint operators : statements:
_tElementary properties of bounded operators ;
_tSpectral theorem for bounded self-adjoint operators, I ;
_tSpectral theorem for bounded self-adjoint operators, II ;
_gExercises --
_tThe spectral theorem for bounded self-adjoint operators : proofs:
_tProof of the spectral theorem, first version ;
_tProof of the spectral theorem, second version ;
_gExercises --
_tUnbounded self-adjoint operators:
_gIntroduction ;
_tAdjoint and closure of an unbounded operator ;
_tElementary properties of adjoints and closed operators ;
_tThe spectrum of an unbounded operator ;
_tConditions for self-adjointness and essential self-adjointness ;
_tA counterexample ;
_tAn example ;
_tThe basic operators of quantum mechanics ;
_tSums of self-adjoint operators ;
_tAnother counterexample ;
_gExercises --
_tThe spectral theorem for unbounded self-adjoint operators:
_tStatements of the spectral theorem ;
_tStone's theorem and one-parameter unitary groups ;
_tThe spectral theorem for bounded normal operators ;
_tProof of the spectral theorem for unbounded self-adjoint operators ;
_gExercises --
_tThe harmonic oscillator:
_tThe role of the harmonic oscillator ;
_tThe algebraic approach ;
_tThe analytic approach ;
_tDomain conditions and completeness ;
_gExercises --
_tThe uncertainty principle:
_tUncertainty principle, first version ;
_tA counterexample ;
_tUncertainty principle, second version ;
_tMinimum uncertainty states ;
_gExercises --
_tQuantization schemes for Euclidean space:
_tOrdering ambiguities ;
_tSome common quantization schemes ;
_tThe Weyl quantization for R²[superscript n] ;
_tThe "No go" theorem of Groenewold ;
_gExercises --
_tThe Stone-Von Neumann theorem:
_tA heuristic argument ;
_tThe exponentiated commutation relations ;
_tThe theorem ;
_tThe Segal-Bargmann space ;
_gExercises --
_tThe WKB approximation:
_gIntroduction ;
_tThe old quantum theory and the Bohr-Sommerfeld condition ;
_tClassical and semiclassical approximations ;
_tThe WKB approximation away from the turning points ;
_tThe Airy function and the connection formulas ;
_tA rigorous error estimate ;
_tOther approaches ;
_gExercises --
_tLie groups, Lie algebras, and representations:
_gSummary ;
_tMatrix Lie groups ;
_tLie algebras ;
_tThe matrix exponential ;
_tThe Lie algebra of a matrix Lie group ;
_tRelationships between Lie groups and Lie algebras ;
_tFinite-dimensional representations of Lie groups and Lie algebras ;
_tNew representations from old ;
_tInfinite-dimensional unitary representations ;
_gExercises --
_tAngular momentum and spin:
_tThe role of angular momentum in quantum mechanics ;
_tThe angular momentum operators in R³ ;
_tAngular momentum from the Lie algebra point of view ;
_tThe irreducible representations of so(3) ;
_tThe irreducible representations of SO(3) ;
_tRealizing the representations inside L²(S²) --
_tRealizing the representations inside L²(M³) ;
_tSpin ;
_tTensor products of representations : "addition of angular momentum" ;
_tVectors and vector operators ;
_gExercises --
_tRadial potentials and the hydrogen atom:
_tRadial potentials ;
_tThe hydrogen atom : preliminaries ;
_tThe bound states of the hydrogen atom ;
_tThe Runge-Lenz vector in the quantum Kepler problem ;
_tThe role of spin ;
_tRunge-Lenz calculations ;
_gExercises --
_tSystems and subsystems, multiple particles:
_gIntroduction ;
_tTrace-class and Hilbert-Schmidt operators ;
_tDensity matrices : the general notion of the state of a quantum system ;
_tModified axioms for quantum mechanics ;
_tComposite systems and the tensor product ;
_tMultiple particles : bosons and fermions ;
_t"Statistics" and the Pauli exclusion principle ;
_gExercises --
_tThe path integral formulation of quantum mechanics:
_tTrotter product formula ;
_tFormal derivation of the Feynman path integral ;
_tThe imaginary-time calculation ;
_tThe Wiener measure ;
_tThe Feynman-Kac formula ;
_tPath integrals in quantum field theory ;
_gExercises --
_tHamiltonian mechanics on manifolds:
_tCalculus on manifolds ;
_tMechanics on symplectic manifolds ;
_gExercises --
_tGeometric quantization on Euclidean space:
_gIntroduction ;
_tPrequantization ;
_tProblems with prequantization ;
_tQuantization ;
_tQuantization of observables ;
_gExercises --
_tGeometric quantization on manifolds:
_gIntroduction ;
_tLine bundles and connections ;
_tPrequantization ;
_tPolarizations ;
_tQuantization without half-forms ;
_tQuantization with half-forms : the real case ;
_tQuantization with half-forms : the complex case ;
_tPairing maps ;
_gExercises --
_tA review of basic material:
_tTensor products of vector spaces ;
_tMeasure theory ;
_tElementary functional analysis ;
_tHilbert spaces and operators on them.
650 0 _aQuantum theory
_xMathematics.
_928451
650 7 _aQuantum theory
_xMathematics.
_2fast
_928451
650 7 _aQuantenmechanik
_2gnd
_928452
650 7 _aMathematische Methode
_2gnd
_928453
830 0 _aGraduate texts in mathematics ;
_v267.
_928454
906 _a7
_bcbc
_cpccadap
_d2
_eepcn
_f20
_gy-gencatlg
942 _2ddc
_cBK
999 _c9845
_d9845