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Prugovecki E. Quantum Mechanics in Hilbert Space
Prugovecki E.  Quantum Mechanics in Hilbert Space







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: Quantum Mechanics in Hilbert Space

: Prugovecki E.

:

A rigorous and critical presentation of the basic mathematics of nonrelativistic quantum mechanics, this text is suitable for courses in functional analysis at the advanced undergraduate and graduate levels. Its readable and self-contained form is accessible even to students without an extensive mathematical background.


: en

: /

:

ed2k: ed2k stats

: 2nd edition

: 1981

: 685

: 13.05.2008

: | | ID
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$\mathcal{T}$ operators      511
Accumulation point      27
Additive class      see Algebra Boolean
Algebra, associative      174
Algebra, boolean      58 60
Algebra, commutative      175
Algebra, generated by observables      276
Algebra, maximally Abelian      328
Almost everywhere      90
Asymptotic completeness      552
Asymptotic condition, physical      455
Asymptotic condition, strong      418
Asymptotic Hilbert space      612
Asymptotic states in classical mechanics      417
Asymptotic states in interaction picture      474
Asymptotic states in multichannel scattering      600
Asymptotic states, incoming      418 455
Asymptotic states, modified      514
Asymptotic states, Moller      418
Asymptotic states, outgoing      418 455
Asymptotic states, physical      457
Asymptotic states, strong      418
Axioms      2
Axioms for spin and statistics      309
Banach space      30
Bauers formula      592
Bessels inequality      38
Bilinear form      137
Borel set      62
Born approximations      556
Born series      557 621
Borns correspondence rule      262 266
Bose Einstein statistics      307
Bosons      353
Bound state      49
Bound state, energy eigenvalues of      49
Bounded variation      465
Canonical commutation relations      329
Canonical commutation relations, irreducible representation      341
Canonical commutation relations, reducible representation      341
Canonical commutation relations, representation of      331
Canonical form      see Operators Spectral
Channel, arrangement      596
Channel, concept of      596 597
Channel, energy      597
Channel, incoming      596
Channel, outgoing      596
Characteristic function      81
Characteristic subspace      227
Characteristic value      see Eigenvalue
Characteristic vector      see Eigenvector
Closed graph theorem      210
Closure      27
Cluster of particles      596
Cluster point      see Accumulation point
Collision, rearrangement      598
Completion      27
Confidence function      410
Configuration representation      352
Confluent hypergeometric equation      165
Confluent hypergeometric function      165
Copenhagen school      see Measurement
Correspondence rule      3 259
Coulomb interaction      164
Cross section, differential      426 436 510
Cross section, total      426 513
Cyclic vector      365
Decomposition of identity      see Spectral function
Deficiency index      223
Deficiency subspace      223
Dense embedding      31
Dense set      27
Density matrix      402
Density operator      391
Diffraction effect      7
Dirac delta function      509
Direct sum of Euclidean spaces      133
Distance function      see Metric
Dynamical law      2
Eigenfunction      491 498
Eigenfunction expansion      499
Eigennumber      491
Eigenvalue      48 226
Eigenvalue equation      47
Eigenvalue, degenerate      49
Eigenvalue, multiplicity of      534
Eigenvalue, nondegenerate      49
Eigenvector      226
Energy shell      510 620
Ensemble, state of      393
Equal a priori probabilities, principle of      394
Equation of motion      3
Equivalence, class      28
Equivalence, physical      293
Equivalence, relation      27
Euclidean spaces      18
Euclidean spaces, complete orthonormal system in      see Euclidean spaces orthonormal
Euclidean spaces, isomorphism of      23
Euclidean spaces, orthogonal basis in      36
Euclidean spaces, tensor products of      137
Euclidean spaces, unitary equivalent      see Euclidean spaces isomorphism
Evanescence      301
expectation value      263 277
Extension Principle      188
Faddeev equations      624 625
Fatous Lemma      287
Fermi Dirac statistics      307
fermions      353
Field      58 (see also Alegebra Boolean)
Fourier coefficient      38
Fragments      596
Fragments, internal energy of      597
Fredholm determinant, modified      527
Fredholm determinant, modified, first minor of      527
Fredholm theory      526
Free states      418
Fubinis Theorem      96
Function, appropriate symmetrization of      312
Function, Riemann integrable      90
Function, spectral      235
Function, square integrable      101
Function, total variation of      467
Functional      see also Vector spaces
Functional, bounded      182
Functional, continuous      182
Functional, linear      182
Gelfand triples      see Hilbert spaces rigged
Greens function      501
Greens function, advanced      503
Greens function, free advanced      524
Greens function, free retarded      524
Greens function, full      502 528
Greens function, full, series expansion of      529
Greens function, partial wave      525
Greens function, retarded      503
Greens function, symmetry properties of      531
Greens operator      501
Ground state      168
Hacks theorem      610
Hahns theorem      237
Hamiltonian      286
Hamiltonian of arrangement channel      599
Hamiltonian, eigenfunction expansion for      500
Hamiltonian, internal free      442
Hamiltonian, time-dependent      292 372
harmonic oscillator      55
Heisenbergs equation      3 294
Heisenbergs picture      293
Heisenbergs uncertainty principle      8
Hellinger and Toeplitz theorem      195
Hilbert spaces      30 32 33 41 109 220
Hilbert spaces, equipped      498
Hilbert spaces, extended      186
Hilbert spaces, Hilbert tensor products of      144 145
Hilbert spaces, isometric transformation of      212
Hilbert spaces, rigged      186
Idempotent      200
Identification operator      613
Incident flux      426
Indicial equation      165
Inertial frame      4
Inner product      18
Integral, Bochner      480
Integral, Lebesgue      90 114
Integral, Riemann Stieltjes      463
Integral, Riemann Stieltjes, cross-iterated      464
Integral, Riemann Stieltjes, improper      464
Integral, spectral      483
Integral, time-ordered      372
Interacting state      420
Interaction picture      298
Intertwining property      see Wave operators
Isometric operators and transformations      212
Isomorphism of Euclidean spaces      23
Katos theorem      367
Kernel, $L^2$      see Kernel Hilbert Schmidt
Kernel, Hilbert Schmidt      526
Kernel, trace class      526
Kummers series      165
Laboratory frame of reference      119
Laguerre functions      170
Laguerre polynomials      170
Laguerre polynomials, generating function of      171
Lebesgue bounded (dominated) convergence theorem      287
Lebesgue intergral      see Integral
Lebesgue monotone convergence theorem      92
Legendre functions, associated      152
Legendre polynomials, associated      167
Limit point      537 (see also Accumulation point)
Limit point, Weyls criterion for      537
Limit, strong      see Operators sequences
Limit, uniform      see Operators sequences
Limit, weak      see Operators sequences
Linear manifold      see Vector spaces
Linear operator      see Operators
Linear space      see Vector spaces
Linear subspace      see Vector subspace
Liouville equation      397
Liouville space      405
Lippmann Schwinger equations for $\mathcal{T}(\zeta)$      511
Lippmann Schwinger equations for eigenfunctions      503 618
Lippmann Schwinger equations in Hilbert space      470
Lippmann Schwinger equations, perturbation solution of      456
Lippmann Schwinger equations, solution-type      472
Lippmann Schwinger equations, two-Hilbert space      615
Mapping, Borel measurable      115
Mapping, isometric      27
Mapping, isomorphism      175
Matrix mechanics      297
Mean value theorem      490
Measurable space      67
Measurable space, simple      81
Measure      58 67
Measure spaces      76
Measure spaces, Cartesian product of      77
Measure, complex      232
Measure, continuous      240
Measure, extension of      70
Measure, finite      67 79 232
Measure, inner      79
Measure, Lebesgue      67 78 79 114
Measure, Lebesgue Stieltjes      79
Measure, lower variation of      239
Measure, outer      71
Measure, positive operator-valued      406
Measure, probability      68
Measure, signed      236 237
Measure, spectral      231
Measure, upper variation      239
Measurement, determinative      6 262 424
Measurement, determinative, Borns correspondence rule for      262
Measurement, preparatory      6 264 266
Measurement, theory of      4 262 264
Metric      25
Metric space      25
Metric space, complete      26
Metric space, completion      27
Metric space, densely embedded      27
Moller wave operators      421 603
Momentum representation      354
Monotone class of sets      63
Monotonic sequence of operators      242
Neighborhood      27
Norm      20
Norm, completeness in      30
Norm, convergence in      30
Norm, Hilbert Schmidt      384
Norm, Rollnik      545
Norm, trace norm      386
Observables      2
Observables, compatible      260 262
Observables, complete set of      316 498
Observables, expectation value      277
Observables, fundamental      270
Operators, adjoint      188
Operators, antilinear      173 (see also Transformations antilinear)
Operators, asymptotically compensating      515
Operators, bound of      178
Operators, bounded      242
Operators, bounded, from above      535
Operators, bounded, from below      535
Operators, bounded, relative to      360
Operators, canonical form      379
Operators, canonically conjugate      329
Operators, closed      191
Operators, compact      375
Operators, complete set of      315
Operators, completely continuous      375
Operators, core      366
Operators, density      see Statistical operator
Operators, essentially self-adjoint      358
Operators, extension      186
Operators, finite-rank      377
Operators, function of      272
Operators, graph of      191
Operators, Hermitian      127
Operators, Hilbert Schmidt      383
Operators, hypermaximal      see Self-adjoint operators
Operators, linear      173 186 192 linear)
Operators, linear, closure of      355
Operators, matrix representation of      175 176
Operators, monotone sequences of      242
Operators, positive definite      381
Operators, projection      see Projector
Operators, relative bound      361
Operators, restriction      186
Operators, self-adjoint      see Self-adjoint operators
Operators, sequences, strong limit of      230 231
Operators, sequences, uniform limit of      230
Operators, sequences, weak limit of      230 231
Operators, symmetric      192
Optical theorem      513
Optical theorem, generalized      513
Orthogonal system      21
Orthonormal basis      36
Parsevals relation      38
Partially isometric operators      440
Paulis exclusion principle      168 307
Phase shifts      497 589 593
Phase space      2
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